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Abstract

Railway systems cause more severe vibration pollution than road systems, and the vibration pollution caused by railway systems has been increasing annually over the past two decades. Despite this, there is limited investigation into the accuracy of train-induced vibration prediction methods. Therefore, the present study investigated the accuracy of the detailed and general procedures proposed by the US Federal Transit Administration (FTA) for predicting train-induced vibrations. The accuracy of these methods was determined by evaluating their predictions against in-situ measurements of ground vibration levels induced by passing trains. The field experiments conducted in this study involved the measurement of train-induced vibration levels, the estimation of line-source transfer mobility (LSTM) by dropping a weight on rail sleepers, the evaluation of force density levels, and the determination of ground LSTM by dropping a weight on the ground. In addition, graphs were plotted to determine the relationships of train-induced vertical vibration levels with train speed, distance from the track centreline, and train length. The study compared measured and predicted vibration levels to assess the accuracy of the FTA’s detailed and general prediction methods. Results indicate that detailed assessments tend to overestimate vibration levels, while an optimized general vibration assessment curve offers a more precise estimate. Additionally, on the basis of the measurement data, the study explored the impact of train length on ground vibration and demonstrated that track ballast can absorb vibration energy along the wave propagation path. A frequency propagation contour plot was also created to analyse vibration energy across different frequencies. These findings enhance the reliability of train-induced vibration prediction methods and provide insights for improving vibration mitigation strategies in urban railway systems.

Keywords

force density level transfer mobility vibration assessment railway train

Introduction

Railways are an efficient and comfortable mode of transportation. However, railway systems are operated on tracks with heavy axle loads, fixed distances between axles, and contact between the train wheel and rail. Consequently, railway systems induce more severe vibration pollution than road systems. The growth of railway networks worldwide has resulted in increasing research regarding railway noise and vibration since the year 2000.^1,2 A previous study¹ indicated that 44% and 31% of stipulated minimum levels of vibration and ground-borne noise were exceeded, with similar problems being prevalent worldwide. Contact between the train wheels and rails is a major source of railway vibrations,³ which are amplified by irregularities in wheels (particularly in wheel roundness) and rail surfaces. Vibration mitigation tracks, for example, floating slab track and high resilient fastener, are used when predicted vibration levels exceed permitted vibration levels.

Thus, predictions of train-induced vibrations are essential during the planning and design of railway systems. Methods for predicting these vibrations can be divided into experimental,^4–7 numerical,^8–12 empirical,^2,5,13 and machine learning-based^14,15 variants. Experimental methods, such as those proposed in, Yoshioka (1997)⁴ involve using an excitation force obtained at a reference site to predict train-induced vibrations at the site of interest; this approach is known as the equivalent excitation force method. In Liu et al, a borehole excitation experiment was conducted to obtain a transfer function, and this function was combined with the excitation force obtained from a train–track model to predict underground train-induced vibrations. Most notably, the vibration prediction method proposed in Nelson and Saurenman (1987), which was developed under a research contract funded by the US Federal Transit Administration (FTA),⁵ has been widely used in several studies.^17–20 Based on this method, Verbraken et al. (2011),²¹ derived analytical equations for detailed vibration assessment and verified these equations by using a coupled finite-element and boundary-element method. The FTA’s prediction method has performed well in validation against empirical data, such as in Richard and Karman (2005),²² which tested it across 13 types of buildings. Similarly, Sadeghi et al. (2019)² investigated the FTA’s general vibration assessment method at three Iranian subway lines and proposed an improved model. In Tao et al. (2022),⁷ the vibration level in buildings was estimated, and the results indicated that the FTA’s general vibration assessment method underestimates the vibration attenuation. Moreover, Cola ç o et al. (2022)²³ verified the accuracy of a numerical–experimental hybrid prediction method through onsite measurements. However, few studies have investigated the accuracy of such methods despite the widespread use of train-induced excitation force data in conjunction with transfer mobility experiments to predict train-induced vibrations. Therefore, the present study investigated the accuracy of the FTA’s detailed and general vibration analysis methods.

Accurately predicting and mitigating train-induced vibrations are crucial tasks in urban infrastructure planning. This study examined the accuracy of the FTA’s methods for predicting train-induced vibrations by using field-measured data; these methods involve identifying force density level (FDL) and line-source transfer mobility (LSTM). Field measurements were conducted to capture train-induced vibration levels at various distances from the track centreline. In addition, dropping weight experiments were repeated at precise locations within the track system to evaluate the stability and reproducibility of FDL measurements. Subsequently, ground LSTM, obtained by dropping a weight on the ground, was used for the prediction procedure. Moreover, this study created a frequency propagation contour plot to analyse the vibration energy under different vibration frequencies at various distances. The measured and predicted vibration levels were compared to determine the prediction error. Additionally, the effect of train length on ground vibration was examined. Then, the track ballast within the far and near tracks can absorb ground vibration energy along the wave propagation path was demonstrated. These findings can be utilised to improve the reliability of train-induced vibration prediction methods and to optimise strategies for mitigating vibrations in urban railway systems.

Vibration assessment method

Detailed vibration analysis

The FTA’s detailed vibration analysis method is usually used to assess train-induced ground vibrations in newly constructed railway systems. The predicted vibration $L_{P r e}$ at different distances d from the track centreline can be obtained by using the following equation⁵

L_{p r e, d} (f) = FDL (f) + {L S T M}_{P r e, d} (f) + C_{b}

(1)

where FDL represents the FDL (dB),

{L S T M}_{P r e, d}

denotes the LSTM of the analysed site (dB), and

C_{b}

is an adjustment factor based on the structure’s foundation. Because this study focused solely on free-field ground vibrations,

C_{b}

was ignored in this study.

The FDL in vibration analysis is analogous to sound power in acoustic measurement, and each type of train has a unique FDL spectrum. However, the FDL cannot be directly measured and must be indirectly derived using the following equation

FDL (f) = L_{V, d} (f) - {L S T M}_{F D L, d} (f)

(2)

where

L_{V, d}

represents the measured train-induced vibration and

{L S T M}_{F D L, d}

denotes the LSTM of the combined track–ground system. The method for deriving the FDL is illustrated in Figure 1. Initially, the train-induced vibration

L_{V, d}

is measured at various distances d from the track centreline (Figure 1(a)). The term

{L S T M}_{F D L, d}

can be obtained by integrating the point-source transfer mobility (PSTM)

{P S T M}_{F D L, d}

, which is identified by dropping a weight on a train wheel–track contact point and measuring the vibration velocity on the ground surface (Figure 1(b)). Subsequently, the FDL is obtained using equation (2).

Figure 1.

Method proposed by the FTA for the estimation of the FDL: (a) measurement of train-induced vibration and (b) transfer mobility experiment conducted on a track.

The objective of the FTA’s detailed vibration analysis method is to predict $L_{p r e, d}$ in new rail construction sites. Therefore, the ground’s transfer mobility must be determined for the specific conditions at hand. The LSTM can be determined through summation for discrete input force systems, such as viaduct systems, and through integration for continuous systems input force systems, such as at-grade or subway systems. In equation (3),^20,24 $f$ indicates the central frequency of one-third octave bands, and $a, b$ represents the integration range on both sides of the measurement line¹⁶

{L S T M}_{T, L} (f) = 10 \log_{10} {\int_{a}^{b} [{P S T M}^{2} (f, d)] d x}

(3)

where a, and b are the hypotenuse of the distance d of Figure 1 and half train length X/2, that is,

\sqrt{d^{2} + {(\frac{- X}{2})}^{2}}

, and

\sqrt{d^{2} + {(\frac{X}{2})}^{2}}

, respectively.

Field experiment and measurement strategy

To meet the limitations and requirements outlined in the FTA manual, a sufficiently broad field located close to railway tracks is required to validate the FTA’s detailed vibration analysis method. Therefore, a field located at 24.782403 N and 121.762192 E in Sicheng, Yilan, Taiwan (Figure 2), was selected as the experimental site in this study. This location is adjacent to a conventional double-track railway system, with approximately 800 m of straight continuous welded rail tracks before and after the measurement line. This straight track section does not contain joints and includes special trackwork within 61 m (200 ft). Additionally, the experiment location comprises loose ground that is primarily composed of silt and clay (Figure 3).

Figure 2.

Measurement field for train-induced vibrations: (a) aerial view of the field, (b) an accelerometer on the ground, (c) PSTM experiment on a ballast sleeper, and (d) PSTM experiment on the ground.

Figure 3.

Soil profile and shear wave speed at the measurement site.

Figure 2(a) displays the ground vibrations induced by passing trains measured at distances of 9, 15, 25, and 35 m from the centreline of the near track by using accelerometers mounted on the ground. The measured vibrations combine both vertical and longitudinal (parallel to the track) directions for each distance. Figure 2(b) displays the accelerometers used in this study. PSTM experiments were conducted at the centre of the ballast sleeper to determine the FDL of the trains, as illustrated in Figure 2(c). The transducers used for measuring ground vibration and the impact force of the dropping weight are also listed in Table 1. In this approach, the stiffness values of the rail and the insulating pad under the rail were neglected, resulting in conservative PSTM. The selected sleepers were positioned at the projection point of the measurement line (Figure 4).

Table 1.

Transducer specifications.

Measurement transducer	Manufacturer	Transducer model	Sensitivity
Acceleration	Wilcoxon sensing technologies	WR 731A	10 V/g
Force sensor	PCB piezotronics	PCB-200C50	22.5 mV / kN

Figure 4.

Layout of three in-situ experiments.

To assess the stability and replicability of the FDL, ${L S T M}_{F D L, d}$ was determined for three sleepers located close to each other. The layout for determining ${L S T M}_{F D L, d}$ is shown in Figure 4. Dropping tests were separately conducted on three sleepers. In addition, as shown in Figure 2(d) a dropping weight test was also conducted on the ground to derive ${L S T M}_{P r e, d} (f)$ . This parameter was then used with the obtained FDL spectra to predict train-induced vibrations, and the predicted vibrations were evaluated against measured train-induced ground vibrations.

The trains selected in this study to obtain observations were EMU3000 trains, which are express electrical multiple units. These trains frequently traverse the measurement site and have been used in commercial operations since 2021. Each coach of this train has a weight of approximately 38 tonnes, the end coaches of the train have a length of 21.35 m, the other coaches of the train have a length of 20.3 m, and the total length of the train is 245.7 m.

Measurement of train-induced vibration

Train-induced vibration at the measurement site

Figure 5 displays the train-induced ground vibration levels under varying train speeds and different distances from the track centreline, encompassing both vertical and longitudinal vibrations. The background vibration velocity was approximately 33–45 VdB. According to the regression curve of distance versus vibration velocity, the train-induced vibrations at positions located 9 m away from the centreline of the near track and 14 m away from the centreline of the far track were 72.5–81.3 and 67.7–78.3 VdB, respectively, under a train speed of 53–119 km/h. When trains pass both tracks, the vibration level positively correlated with the train speed. The average regression slope factor for the relationship between train-induced vibrations and train speed for the near and far track was 0.07 and 0.10 VdB, respectively, indicating a notable difference in both slope and vibration levels. However, for the nearest measured distance from both tracks, the slopes were 0.11 and 0.10 equation (3) VdB, respectively, aligning with the FTA’s proposed train speed adjustment equation,⁵ which is expressed as follows

{A d j}_{s p e e d} (d B) = 20 \log (\frac{V}{V_{r e f}})

(4)

where

{A d j}_{s p e e d}

is the difference in vibration level between train speed

V

and the reference train speed

V_{r e f}

Figure 5.

Train-induced vibration velocity under varying train speeds at different distances from the track centreline: (a) near and (b) far track.

Additionally, the integration approach used to obtain LSTM involves setting the integration bounds to the length of the train.¹⁶ Research suggests a positive correlation between ground vibration and train length under constant conditions. Figure 6 illustrates the relationships obtained between ground vibration levels and train speed for different train lengths (4–14 cars) during field investigations of conventional railways. Figures 6(a) and (b) indicate that among all investigated trains, most of the four-car commuter trains induced the lowest vibration levels at positions located 9 m from the centreline of the near track and 14 m from the centreline of the far track under a train speed of approximately 58 to 97 km/h, respectively. However, no clear positive correlation was observed between train length and measured vibration levels for longer trains.

Figure 6.

Vibrations induced by trains with different numbers of cars under varying train speeds: (a) at a position 9 m from the centreline of the near track and (b) at a position 14 m from the centreline of the far track.

Spectrum of the train-induced vibration

This study conducted repeated field measurements to validate the FTA’s method for assessing train-induced ground vibration. Figures 7(a)–(d) illustrates the vertical ground vibration spectra induced by EMU3000 trains at positions located 9, 15, 25, and 35 m from the centreline of the near track, respectively. Figure 7(e) shows the standard deviation of each frequency band of EMU3000-induced vibrations at the aforementioned positions. Most EMU3000 trains that passed through the experiment location had a speed of 105 ± 5 km/h, resulting in minor differences in the induced ground vibrations due to speed variations. Figure 7 indicates that the trend of train-induced ground vibration level was highly consistent when the vibration frequency was above 8 Hz at each position. The maximum standard deviations in the vibration levels at positions located 9, 15, 25, and 35 m away from the centreline of the near track were 4.3, 3.9, 3.9, and 3.2 dB, respectively. Moreover, the dominant vibration frequency at this measurement site was between 10 and 12.5 Hz.

Figure 7.

(a)–(d) Ground vertical vibration spectrum induced by 12-car EMU3000 trains at different distances from the centreline of the near track: (a) 9, (b) 15, (c) 25, and (d) 35 m. (e) Standard deviations of the vibration levels at 9, 15, 25, and 35 m from the track centreline.

The vertical vibration energy decreased with distance from the track centreline (Figure 7). The energy decay was negligible for frequencies below 4 Hz, moderate for frequencies between 4 and 31.5 Hz, and significant for frequencies above 31.5 Hz, particularly between 9 and 15 m from the track centreline. The spectrum of train-induced vibrations and the energy decay pattern depend on the ground composition,²⁵ and the observations of the present study were consistent with the pattern of vibration energy decay on soft ground. Nevertheless, the spectrum decay rates corresponding to different soil compositions should be further investigated in the future.

FDL and LSTM

Determining LSTM by dropping a weight on ballast track sleepers

After the train-induced ground vibration level and vibration spectra were obtained through direct measurement, the detailed vibration analysis method of the FTA was applied. The FDL must be derived before this method is used. First, the measured $L_{V, d}$ spectra (Figure 7) were substituted into equation (2) to derive the FDLs of the investigated trains. Subsequently, the line-source mobility of the combined track–ground system ( ${L S T M}_{F D L, d}$ ) was determined by integrating the PSTM ( ${P S T M}_{F D L, d}$ ) obtained through in-situ experiments where weights were dropped on ballast track sleepers, as shown in Figure 2(c). According to Nelson and Saurenman,¹⁶ the PSTM should be integrated over the train length to obtain the LSTM; therefore, the integrated length was set as 245.6 m (the length of EMU3000) in the present study.

The obtained ${L S T M}_{F D L, d}$ spectra are shown in Figure 8. In this figure, LSTM spectra for each individual train are represented as lightly coloured lines, while the average LSTM spectra for each train are depicted by lines with circular dot. In Figure 8, variations in LSTM were inconsistent for frequencies below 8 Hz; however, when the frequency exceeded 8 Hz, the LSTM spectra concentrated within a narrower range. This is because the duration of the measured impact force was typically less than 10 ms, which limits the time-frequency transfer and causes instability in the LSTM spectra at lower frequencies. These findings align with the results obtained in US Federal Transit Administration (FTA), where the LSTM spectra were only presented for a vibration frequency range of 6–400 Hz. Additionally, ${L S T M}_{F D L, d}$ spectra was dominant for frequencies between 10 and 16 Hz. Considerable differences were observed in the ${L S T M}_{F D L, d}$ magnitudes obtained through experiments where weights were dropped on three sleepers (Figure 4). For instance, the ${L S T M}_{F D L, 9 m}$ magnitudes had a maximum difference of 10.9 and 12 dB at 16 and 80 Hz, respectively. Figure 9 depicts the standard deviations of the LSTM spectra obtained at various distances from the point at which the weight landed on the sleepers. The findings in the figure reveal that as the distance from the impact point increased, variations in LSTM became less uniform. In conclusion, when the vibration frequency was above 8 Hz, ${L S T M}_{F D L, d}$ had maximum standard deviations of 2.1, 3.0, and 1.5 dB in the three experiments.

Figure 8.

LSTM obtained through experiments involving the dropping of a weight on ballast track sleepers: LSTM for positions located at (a) 9, (b) 15, (c) 25, and (d) 35 m from the centreline of the near track.

Figure 9.

Standard deviations of the LSTM obtained through experiments involving the dropping of a weight on ballast track sleepers: (a) first measurement, (b) second measurement, and (3) third measurement.

Derivation of FDL

The terms $L_{V, d}$ and ${L S T M}_{F D L, d}$ were derived through repeated measurements at a single location. The FDL spectra of EMU3000 at different distances from the track centreline in each experiment are depicted in Figures 10(a)–(c). The FDL was nearly constant within a frequency range of 4–31.5 Hz; however, the FDL spectra in different experiments at the same distance from the track centreline exhibited noticeable variations. On the basis of the relevant literature and experimental experience, this variation is attributed to two major reasons. First, $L_{V, d}$ can result in train wheels losing their roundness, variations in vehicle suspension conditions, and rail irregularities.²⁶ Second, each sleeper has different coupling conditions with the ballast surrounding it, which results in variations in ${L S T M}_{F D L, d}$ when a weight is dropped on the sleepers. The deviations in $L_{V, d}$ and ${L S T M}_{F D L, d}$ are shown in Figures 7 and 9, respectively. Furthermore, Figure 11 displays the standard deviations in the FDL of EMU3000 over three experiments. When the vibration frequency exceeded 8 Hz, the maximum standard deviations at distances of 9, 15, 25, and 35 m from the centreline of the near track were 5.5, 5.2, 7.5, and 6.0 dB, respectively. These FDL variations notably influence the predicted ground vibration and merit further investigation.

Figure 10.

FDL spectra in three experiments each at different locations: (a) first, (b) second, and (c) third experiments.

Figure 11.

Standard deviations of FDL in three experiments each at different locations: (a) first, (b) second, and (c) third experiments.

Ground LSTM at the experiment location

The term ${P S T M}_{P r e, d}$ was experimentally determined (Figure 2) by dropping a weight on the ground. This experiment measured the decay rate of each frequency band at distances of 6, 12, 20, and 27 m from the point of impact. Figure 12 displays the ${L S T M}_{P r e, d}$ spectra obtained by integrating the experimentally determined ${P S T M}_{P r e, d}$ over half the train length on both sides of the measurement line using equation (3). The individual and average ${L S T M}_{P r e, d}$ spectra at four distances from the track centreline are displayed in Figure 12(a). For each distance, the maximum ${L S T M}_{P r e, d}$ magnitude was observed at 25 Hz. This value decreased by 0.31 dB with each unit increase in distance. Similar to ${L S T M}_{F D L, d}$ , ${L S T M}_{P r e, d}$ also exhibited variations when the vibration frequency was below 8 Hz of each individual drop test. Figure 12(b) shows the standard deviations of ${L S T M}_{P r e, d}$ , and by comparing it with ${L S T M}_{F D L, d}$ , ${L S T M}_{P r e, d}$ exhibits more consistent results. When the vibration frequency was above 8 Hz, the maximum standard deviation of ${L S T M}_{P r e, d}$ was 0.9, 0.9, 1.0, and 1.1 dB at distances located 6, 12, 20, and 27 m from the track centreline, respectively; thus, the standard deviations in ${L S T M}_{P r e, d}$ were considerably lower than those in the FDL. Moreover, as the distance from the track centreline increased, the standard deviations of ${L S T M}_{P r e, d}$ also increased, which is similar to the results reported in Ref. 6. This finding is attributed to reduced coherence between the input force and the output acceleration with increasing distance.

Figure 12.

Ground LSTM spectra: (a) LSTM spectra obtained for each test and (b) standard deviations of LSTM.

Comparison of prediction and measurement results

After FDL and ${L S T M}_{P r e, d}$ spectra are obtained, the predicted train-induced vibration $L_{p r e d, d}$ can be derived using equation (1). Subsequently, the predicted and measured train-induced ground vibration spectra can be compared. Figure 13 illustrates the variation in the FDL with the distance from the track centreline of the near track. The FDLs at four distances from the track centreline in each experiment were used to predict ground vibration spectra. In Figure 14, the predicted ground vibration spectra are indicated by coloured lines, while black lines represent the measured EMU3000-induced ground vibration average spectra. Figure 14 indicates that the predicted frequencies were mostly higher than the measured ones, with the maximum difference being observed in the range of 20–50 Hz; the maximum differences at positions located 9, 15, 25, and 35 m from the centreline of the near track were 18.3, 17.9, 19.3, and 17.9 dB, respectively (Figure 15). Thus, considerable differences were observed between the predicted and measured vibration spectra.

Figure 13.

Average FDL spectra over three experiments at positions located 9, 15, 25, and 35 m from the centreline of the near track.

Figure 14.

Comparison of the measured and predicted train-induced vertical vibration levels at different distances from the centreline of the near track: (a) 9, (b) 15, (c) 25, and (d) 35 m from the track centreline.

Figure 15.

Differences between the predicted and measured train-induced vertical vibration spectra: (a) 9, (b) 15, (c) 25, and (d) 35 m from the centreline of the near track.

Major parameters influencing the vibration predictions

At-grade track vibration propagation and validation of general vibration assessment

A comparison of Figures 5(a) and (b) indicated that the vibration level did not strictly correspond to the distance from the centreline of the tracks. The distinction between the near and far tracks also influences the train-induced vibration trend. Figure 16 depicts the train-induced vibrations (combining vertical and longitudinal directions) at different distances from the near and far tracks under different train speeds. It is evident, that the vibration level of the near track (circular dots in Figure 16) was higher than that of the far track (square dots in Figure 16), even when the distance from the centreline of the near track was greater than that from the centreline of the far track. Therefore, the general vibration assessment method proposed by the FTA represented by equations (5), and (6) will need to be optimised when applied to multiple-track system. Equation (5) is the locomotive-powered passenger curve (L-curve), and equation (6) is the rapid transit/light rail vehicle curve (R-curve). These curves provide the following relationships between distance from the track centreline and vibration level

L_{v} = 92.28 + 14.81 \log (d) - 14.17 \log {(d)}^{2} + 1.65 \log {(d)}^{3}

(5)

R_{v} = 85.88 - 1.06 \log (d) - 2.32 \log {(d)}^{2} - 0.87 \log {(d)}^{3}

(6)

where d indicates the distance from the track centreline.

Figure 16.

Comparison of general vibration assessment and train-induced vibration levels at different distances from the near and far tracks under different train speeds.

The aforementioned curves, based on a reference train speed of 80.5 km/h (50 mph), were adjusted for a train speed of 50 and 140 km/h by using equation (4). Figure 16 shows the vibration levels obtained through the general vibration assessment method were different from the measured levels, which is similar to the results obtained in Ref. 2. Among the two curves, the R-curve was closer to the measured results of the far track but was unconservative for the near track of this measurement site. Consequently, the solid blue line in represents the optimised curve for the near track vibration levels

L_{v} = 92.28 + 14.81 \log (d) - 14.17 \log {(d)}^{2}

(7)

which aligns well with the measured train-induced vibration levels at different distances from the near track. It can also be observed that the vibration levels induced at the far track were generally lower than those at the near track at this measurement site, and comparing the R-curve to the optimised R-curve reveals that the difference increases with distance. This finding suggests that the vibration energy induced by passing trains is predominantly carried by Rayleigh waves on the ground surface.^5,27 Figure 17 compares the one-third octave bands of ground vibration induced by 96 passing trains at the 9/14 m (from the near/far track centreline to the 1^st measurement point) and 15/20 m (from the near/far track centreline to the 2^nd measurement point) measurement point by vertical and longitudinal directions, respectively. It is clear that at this measurement site, when the spectrum is below 12.5 Hz, both vertical and longitudinal ground vibrations exhibit a similar trend for both tracks, whereas, above 12.5 Hz, the far track shows significantly lower vibration magnitudes than the near track at each distance. Comparing the one-third octave bands at 9/14 m and 15/20 m, as shown in Figure 17, reveals that at this measurement site, the track ballast within the near and far tracks absorbs high-frequency ground vibration energy along the wave propagation path, influencing the ground vibration propagation trends. Consequently, high-frequency vibrations from the near track contribute more significantly to the overall vibration energy than those from the far track. Further research is needed to investigate the differences in attenuation on both sides of a multiple ballast track system.

Figure 17.

Comparison of train-induced vibration frequency bands between the near track and far track (a) vertical direction at 9/14 m (b) longitudinal direction at 9/14 m (c) vertical direction at 15/20 m (d) longitudinal direction at 15/20 m.

Frequency propagation contour plot

The detailed vibration analysis method of the FTA requires the FDL and ${L S T M}_{P r e, d}$ to be determined for the analysed area, as shown in equation (1). Moreover, in equation (2), ${L S T M}_{F D L}$ must be determined to derive the FDL. Thus, the LSTM plays a vital role in the detailed vibration analysis method. To analyse the energy propagation trends across specific frequencies, a frequency propagation contour plot based on integration-determined LSTM spectra was created. Initially, the length of integration in equation (3) for calculating the LSTM was set to range from 2 and 200 m. These spectra are denoted as $LSTM (X, f)$ , where X and f represent the integration length and central frequency of one-third octave bands, respectively. To standardise the integrated LSTM across different integration lengths, each frequency was normalised by dividing it by the frequency magnitude for an integration length of 200 m. This normalised ratio effectively represents the energy propagation under each frequency. Figure 18 displays the frequency propagation contour plot produced in this study. This plot depicts frequencies, distances from the centreline of the relevant tracks, and energy propagation ratios. When the aforementioned normalised ratio approached 1, frequency bands converged, which indicated that minimal additional energy was produced beyond the corresponding distance. Observations at the measurement site (Figure 18) indicated that 90% of the convergence value for frequencies above 4 Hz was achieved when the integration length exceeded approximately 30 m. The convergence for frequencies above 8 Hz was 98% when the integration length was between 50 and 75 m. The frequency propagation contour plot enhances our understanding of the vibration energy under different vibration frequencies at a certain site. Moreover, this plot offers valuable insights for the development of effective vibration mitigation measures. For example, the plot can be used as a reference to identify vibration frequency propagation length, thus enabling the appropriate design of the layout length of a vibration mitigation track.

Figure 18.

Frequency propagation contour plot.

Furthermore, Figure 18 offers insights into the influence of integration length. For instance, when the train length was less than 80 m, the vibration energy showed significant sensitivity to variations in train length. Conversely, for longer train lengths, energy propagation tended to converge, indicating reduced sensitivity of vibration energy to train length. These observations align with the results in Figure 6, which demonstrate that train length did not correlate positively with vibration intensity for trains with more than four-car. To further confirm the impact of train length on vibration energy propagation, Figures 19(a)–(c) present in-situ data of train-induced vertical vibrations, including short-time Fourier transform spectrum and vertical vibration history data for 12-car express train (EMU3000), eight-car tilting train (TEMU2000), and four-car commuter train (EMU500), respectively. In these figures, the red lines denote the arrival and departure times of the first and last axles at the measurement line, whereas the grey lines indicate the time 2 s before and after axle passage. Figure 19 confirms that the train-induced vibrations dissipated to an undetectable level 2 s before and after each train passed, which are approximately 54.5, 60.1, and 49.2 m away from the measurement line.

Figure 19.

Short-time Fourier transform spectrum and vertical vibration history data for (a) an EMU3000 12-car express train, (b) a TEMU2000 eight-car tilting train, and (c) an EMU500 four-car commuter train.

Comparison of predicted and measured train-induced vibration levels

Figure 20(a) displays the measured and predicted train-induced vibration levels at various distances from the centreline of the near track. Under a train speed of approximately 100 km/h, the attenuation in the vertical vibration level with the distance from the centreline of the near track was minimal at low vibration frequencies. The predicted ground vibration levels, which were obtained from the average FDL spectra obtained from three experiments, exhibited larger attenuation with distance than the measured ground vibration levels. This discrepancy is primarily due to the standard deviation of the FDL. The variation in the FDL was influenced by $L_{V, d}$ and ${L S T M}_{F D L, d}$ . The term $L_{V, d}$ varied considerably across different train and rail conditions, and ${L S T M}_{F D L, d}$ was affected by sleeper and ballast coupling conditions (Figure 9).

Figure 20.

Comparison of the measured and predicted train-induced vibrations: (a) vibration spectrum comparison and (b) vibration level comparison.

To further evaluate the prediction results, the train-induced vibration levels were determined through the general vibration assessment method of the FTA, which involves plotting an R-curve and an optimised R-curve by using equations (6), and (7), respectively. Figure 20(b) displays the train-induced vibration levels determined using the FTA’s detailed and general vibration assessment methods. The detailed vibration analysis method overestimated the vibration level at distances of 9–35 m from the centreline of the near track by 12.5–2.7 VdB, even though it only considered vertical vibrations. The degree of overestimation decreased at distances further from the centreline. The vibration levels predicted by R-curve and optimised R-curve, at distances of 9–35 m from the centreline of the near track differed from the corresponding measured vibration levels by 1.8–5.7 VdB and by 4.6–1.9 VdB, respectively. Figure 20 shows that the detailed vibration assessment tends to overestimate the vibration level, whereas the optimised R-curve provides a more precise and conservative estimation of vibration levels. The aforementioned validation was conducted at a single site; therefore, further validation should involve multiple sites.

Conclusions

This study utilized field measurements of train-induced vibration levels, LSTM, and FDL to validate the FTA’s general and detailed methods for predicting train-induced vibrations. Train-induced vibrations were analysed under different train speeds, distances, and lengths. On the basis of the measurement data, a frequency propagation contour plot was created to analyse the vibration energy under different vibration frequencies at various distances. The measured and predicted vibration levels were compared to determine the accuracy of the prediction methods. This study has the following main findings.

1. The FTA’s train speed adjustment equation was validated for the closest measured distances from both tracks, with slopes of 0.11 and 0.10 VdB, respectively.

2. The ground vibration attenuation trends caused by trains passing over the near and far tracks differ at this measurement site. These observations indicate that high-frequency vibrations from the near track contribute more significantly to the overall vibration energy than those from the far track. Further research is needed to investigate the differences in attenuation on both sides of a multiple ballast track system.

3. Regarding the FTA detailed vibration analysis method, FDL standard deviations varied significantly, reaching 5.5–7.5 dB at distances away from the centreline of the near track. These standard deviations were influenced by variations in $L_{V, d}$ and ${L S T M}_{F D L, d}$ . Factors such as train wheel roundness, rail irregularities, and sleeper-ballast coupling conditions, which are not eliminable or addressable by the existing test, contributed to these variations. Additionally, instability in the LSTM spectra at lower frequencies further amplified the FDL standard deviations. These discrepancies indicate potential for refinement in methods used to determine the FDL in future research.

4. The maximum differences between the measurement and FTA detailed vibration analysis predicted spectra ranged from 18.3 to 19.3 dB at distances from the near track centreline, specifically within 20–50 Hz. Additionally, the detailed vibration analysis method overestimated total vibration levels by 12.5–2.7 dB at 9–35 m distances from the near track centreline. On the other hand, the FTA’s general vibration assessment method yielded more accurate predictions after optimising the R-curve, with differences ranging from 4.6 to 1.9 VdB compared to measurements, offering a more precise estimation.

5. This study generated a frequency propagation contour plot to understand energy propagation trends under specific vibration frequencies, aiding in developing effective mitigation measures. The measurement results indicated that the four-car commuter train (shortest train) produced the lowest vertical vibration levels. However, no discernible clear correlation existed between train length and vibration intensity for longer trains.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: this work was supported by the National Science and Technology Council; NSTC 112-2221-E-019-019; NSTC 113-2221-E-019-021.

ORCID iD

Wei-Lun Hsu

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Comparative analysis of methods for predicting train-induced vibrations

Abstract

Keywords

Introduction

Vibration assessment method

Detailed vibration analysis

Field experiment and measurement strategy

Measurement of train-induced vibration

Train-induced vibration at the measurement site

Spectrum of the train-induced vibration

FDL and LSTM

Determining LSTM by dropping a weight on ballast track sleepers

Derivation of FDL

Ground LSTM at the experiment location

Comparison of prediction and measurement results

Major parameters influencing the vibration predictions

At-grade track vibration propagation and validation of general vibration assessment

Frequency propagation contour plot

Comparison of predicted and measured train-induced vibration levels

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References