Identification of line-source transfer mobility and force density by ground acceleration measurements

Ground-borne vibration caused by railway traffic has been a research concern due to its possible side effects on nearby residences. The force density and line-source mobility can effectively characterize the generation and transmission of train-induced vibrations, respectively. This research proposed a frequency-domain method for identifying the line-source transfer mobility and force density using measured vibrations at the ground surface, which was on the basis of the least-square method. The proposed method was applied to a case study at Shenzhen Metro in China, where a total of seven fixed-point hammer impacts with 3.3 m equal intervals were used to represent the train vibration excitations. Line-source transfer mobility of the site and force density levels of the metro train were identified, respectively. Causes for different dominant frequencies can be traced by separating the dynamic characteristics of vibration excitation and transmission. It was found in the case study that at a location 3 m away from the track, the peak at 50 Hz was caused by excitations, while that at 63 Hz was attributed to transmission efficiency related to the soil properties. Subsequently, numerical validations of the fixed-point loads’ assumption and identified force density levels were carried out. Good comparisons between numerically predicted and experimentally identified force density levels indicated the feasibility of the proposed method. At last, the identified line-source transfer mobility and force density levels were applied to the forward problem, i.e., making predictions of train-induced vibrations. The predicted ground and structural vibrations at different locations were compared to corresponding measurements, with good agreement, which experimentally validated the identification method. The identification results of the case study can be employed by similar railway systems as a good reference.


Introduction
The urban railway system plays an important role in the transportation of megacities. With the increase in train speed and loads, as well as the decrease in the distance between railway tracks and residences, the environmental vibration induced by train operations gradually becomes a global research focus in recent decades because of its possible side effects on the serviceability of nearby buildings (Connolly et al. 2016;He et al. 2022a, b).
Vast experimental (Sanayei et al. 2014;Tao et al. 2019;Luo et al. 2022;Jin et al. 2022Jin et al. , 2023 and numerical (Bucinskas et al. 2021;Tao et al. 2021;Xu and Ma 2023) studies on traininduced vibrations have been conducted to understand the generation and propagation mechanisms. General rules are found consistent by different field measurements, such as traininduced vibration levels increase with train speed and loads, and the train-induced vibration spectra are case-dependent and determined by dynamic characteristics of the track structure and site soil (Takemiya 2008). Although the field experimental study is the most direct way to characterize and assess traininduced vibrations, it is costly to obtain and cannot offer data support prior to construction. The numerical and empirical prediction methods are therefore supplementary and helpful in assessments of vibration levels, as well as designs of vibration control measures. According to the vibration generation and transmission mechanisms, there are three primary related research objects (Connolly et al. 2015; Thompson et al. 2019), i.e., the train-track subsystem (excitation), the ground subsystem (transmission path), and the building subsystem (vibration receiver). The widely used empirical approach (Hanson et al. 2012;Quagliata et al. 2018;Sadeghi et al. 2019) assumes that the dynamic characteristics of train-induced vibration excitation and transmission are separable and can be described using force density and line-source transfer mobility, respectively. Fully three-dimensional coupled dynamic models for predicting train-induced vibration are few due to the involved physical complexity and formidable computational cost. To make the prediction model practical in use, certain simplifications and assumptions are indispensable. The substructure method is generally used in numerical models, through which the dynamics of each aforementioned subsystem are investigated separately. The train-track dynamically coupled models of different degrees of detail are the most frequently used for simulating the excitation subsystem (Zhai et al. 2009;Liu et al. 2022). The finite element method (Gao et al. 2019), boundary element method (Costa et al. 2012), and thin-layer stiffness method (Jones et al. 1998) are prevalent in modeling train-induced vibration propagations within the soil. The finite element method (Tao et al. 2021) and impedance method (Sanayei et al. 2011;Zou et al. 2022) can be applied directly to predict traininduced vibration transmissions within the building structures given vibration inputs at the base. More recently, dynamic couplings between subsystems are investigated in detail (Tao et al. 2023) to stress the vibration generation and propagation in a systematic view. Although plenty of investigations have been done on developing numerical models for train-induced vibration predictions, the parameters included are hard to determine and include uncertainties (Liu et al. 2023a).
Given concerns about the disadvantages of field measurements and numerical models mentioned above, hybrid methods of combing field measurements and prediction models gain increasing attraction (Li et al. 2023). To alleviate the challenges faced by modeling or measuring train excitations, Uhl (2007) proposed a method for identifying wheel-rail contact force based on measurements of axle box accelerations and system transfer functions. Lee et al. (2012) proposed a two-step methodology for predicting train-induced ground-borne vibrations. The first step was the back estimation of the effective train loads acting on the bridge piers using field measurements and the 2D FEM. The second step was applying the identified train excitations from the previous step to the 2D FEM for predicting train-induced ground-borne vibrations at other locations. This method aims to avoid the train-track-bridge coupled modeling for obtaining train excitations. Li et al. (2021) and Liu et al. (2023b) followed this two-step (back-estimation and forwardprediction) framework for predicting train-induced groundborne vibrations. In their studies, the vertical wheel-rail contact forces were first identified based on measured rail accelerations and the proposed periodic analytical track model, which were then transferred to excitation forces acting on the subgrade and applied to a 3D FEM for predicting underground metro train induced ground-borne vibrations.
Inspired by the fast prediction needs in practice and the researchdeveloping tendency towards the hybrid method, a frequencydomain method ("Identification method for force density and line transfer mobility" section) is proposed in this research for identifying the line-source transfer mobility and force density using measured vibrations at two locations on the ground surface, where the identified results can be used straightforwardly for predicting train-induced ground-borne vibrations. It was applied to a case study ("Field measurements" section) in Shenzhen metro in China. The identification results ("Identification of line-source transfer mobility" and "Identification of force density levels for metro train" sections) were validated numerically ("Numerical validations of force density levels" section) and experimentally ("Experimental validations of predicted train-induced vibrations" section). Although the proposed method is a hybrid of field measurements and empirical prediction formula, the required measurements are easy to obtain, and it can achieve fast prediction by avoiding sophisticated numerical modeling. The limitations of the proposed method lie in it being more suitable for the near fields because of the poor signal-to-noise ratio of measurements farther away.

Train-induced vibration prediction formula
The US federal transit administration (Quagliata et al. 2018) had reported a detailed train-induced ground-borne vibration prediction procedure, whose formula was shown in Eq. (1).
where L Vtrain dB(R, ) is the train-induced ground surface velocity levels (in decibel) at the receiver location designated as R., L Ftrain dB(P, ) is the force density levels (in decibel) representing the train excitations exerted on a series of positions collected in , and TM L dB(P, R, ) is the line-source transfer mobility levels (in decibel) characterizing wave transmission properties from vibration excitation locations to the ground receiver location R. The reference values for ground surface velocity level, force density level, and transfer mobility level in this research are 1 × 10 −8 m∕s , 1N∕ √ m , and 1 × 10 −8 (m∕s) The force density is defined as the force per root distance along the track (Quagliata et al. 2018), which reflects the wheel-rail contact force exciting the ground. The line-source transfer mobility is an energy summation from point-source transfer mobilities, which can be expressed as: where TM p P j , R, is the point-source transfer mobility from the j th impact location P j at the track to the ground surface receiver location R. It equals the complex velocity response (1) L Vtrain dB(R, ) = L Ftrain dB(P, ) + TM L dB(P, R, ) at the receiver location R caused by a unit point load at P j as a function of frequency (Quagliata et al. 2018). In Eq.
(2), h is the distance interval (m) between adjacent impacts at the track. TM ref is the transfer mobility reference value. The basic assumption behind Eq. (1) is that the dynamic characteristics of the vibration source and transmission path are independent of each other. Therefore, the force density is only dependent on train and track configurations, while the source transfer mobility is intrinsically determined by the wave propagation characteristics of the supporting ground (Lei 2020). It is therefore significant that force density level L Ftrain dB( , ) for different types of trains and tracks is investigated and can be referred to for vibration predictions of future similar railway systems.

Construction of objective function and bounds
The force density level of a train passby can be estimated based on Eq. (1) by subtracting the line-source transfer mobility levels from the train-induced vibration levels at a ground surface receiver location. The line-source transfer mobility levels can be estimated with field hammer impacting measurements (Kuo et al. 2019) based on Eq. (2). The plan view of the general field hammer impacting test is shown in Fig. 1, where a total of n equidistant vertical impacts are exerted at the railhead.
According to Eqs. (1) and (2), the force density levels of a train can be obtained with measurements of train-induced velocities and hammer impacting induced transfer mobilities. The force density level of a train identified from synchronized measurements at two different receiver locations should be the same since it only reflects the vibration excitation characteristics, which is depicted in Eq. (3). If the hammer impacting forces were unknown, the point-source transfer mobilities TM p P j , R, will also be unknown functions of frequency. Nevertheless, through the construction of the objective function (Lee et al. 2012) shown in Eq. (4), the unknowns can be solved using optimization algorithms for multiple parameters such as genetic algorithm.
In Eq. (4), is the given frequency in rad/s, L V P j , R, is the velocity (m/s) at the receiver location R caused by a point load exerted at the location P j , and L F P j , is the point load amplitude (N) acted at the location P j .
The nonlinear objective function formulated in Eq. (4) is subject to bounds on the parameters as: The lower bound L Fl ( ) and upper bound L Fu ( ) in Eq. (5) are deduced from a reasonable range L Ftrainl dB(P, ), L Ftrainu dB( , ) of the force density level of a train. The lower and upper bounds in decibels of the force density levels of a train can be referred to the FTA guideline (Quagliata et al. 2018), as well as other published papers (Kuo et al. 2016;Rajaram and Saurenman 2015). The force density level range is selected to be 0 to 50 dB in this research for a metro train. It is worth noting that derivations in this section are performed in frequency domain, and force density levels and velocity levels are both frequency dependent. The expressions for lower and upper bounds shown in Eq. (7) are therefore also frequency dependent. (3) ,

Optimization procedure based on genetic algorithm
There were a bunch of algorithms for solving the optimization problem, such as genetic algorithms, evolutionary programming, evolution strategies, and simulated annealing (Koh et al. 2000). As derived in the "Construction of objective function and bounds" section, the proposed method only needs measurements at two locations for constructing the objective function, which is optimized to estimate the hammer force amplitudes. Subsequently, the estimated hammer forces are used to identify the line-source transfer mobility for the soil and force density for the train, which can be used in the forward prediction step to predict train-induced ground and structural vibrations at different locations. To achieve the purpose of estimating the hammer force amplitudes in the objective function, i.e., Eq. (4), the genetic algorithm was used in this research due to its global searching capability and no requirement on derivatives.
The optimization approach based on genetic algorithm (GA) is inspired by Darwin's theory of evolution and attempts to imitate natural evolution (Bajpai and Kumar 2010). A set of parameters is called a chromosome in the genetic algorithm, which represents a possible solution to the given problem. Multiple chromosomes or individuals make up a population located within the search space. Procedures of selection, reproduction, crossover, and mutation are then took place within a population, which results in a new population with more favorable offspring. The imitation procedure of natural selection in genetic algorithm runs iteratively until the end condition is met, such as the maximum limit of generations or objective function tolerance. Figure 2 is the flowchart of the proposed method encoded and implemented in MATLAB.

Field measurements
Field measurements were carried out at Qianhai metro depot in Shenzhen, China (Tao et al. 2019). Figure 3a shows a photo of tracks at the throat area of Qianhai depot, as well as the rail for hammer-impacting experiments. Figure 3b is the simplified horizontal layered soil profile of the measurement site. The properties of each soil layer are listed in Table 1. The track configurations in the throat area of a depot are complicated due to the existence of rail joints, curvature, switches, etc. Numerical solutions for predicting train-induced vibrations above the throat area need to simplify the vibration excitation (Tao et al. 2021). The challenges of numerical modeling of such a complex train excitation system and deep site soils intrigue this research on identifications of force density levels and transfer mobility based on field measurements and the FTA-based formula.
The plan view of field setups is referred to Fig. 1, where a total of seven locations (n = 7) with 3.3 m equal intervals (h = 3.3 m) at the rail head were impacted with the steel hammer. Accelerometers were mounted at the ground surface and platform columns for Loc 1 and Loc 2 which were 3 m (L1 = 3 m) and 9 m (L2 = 6 m) away from the track, respectively. Both train-induced and hammer-induced ground and structural vibrations at Loc 1 and Loc 2 were collected. The sampling frequency is 1024 Hz.
The metro train operated at Qianhai metro depot consists of 6 cars. For the type of metro train operated at Qianhai depot, the wheelbase of a bogie is 2.5 m and the distance between two bogies in a car is 15.7 m. The distance between the first and the last axle within a car is approximately   Figure 4 shows the time history of train-induced vibrations at the ground surface located 3 m away from the track during one train passage, where the pass-by moments of 6 cars and 12 bogies can be obviously distinguished. Figure 5a shows the measured ground acceleration time histories at Loc 1 induced by the nearest hammer impact #1 and farthest hammer impact #7. Figure 5b displayed the corresponding velocity levels in one-third octave band of the impact induced ground vibrations shown in Fig. 5a. It is seen that the ground velocity levels induced by hammer impacts are reliable above 12.5 Hz since they exceed the ambient vibrations by more than 10 dB in this frequency range. Therefore, a frequency range from 12.5 to 100 Hz will be adopted for subsequent analysis in this research both considering the field measurement data reliability and upperfrequency limit in the FTA guideline (Quagliata et al. 2018).

Identification of line-source transfer mobility
This section displayed the identified line-source transfer mobility which is a summation from point-source transfer mobilities induced by the hammer impacts exerted on the railhead. It characterizes dynamic properties of vibration propagation within the soil. To obtain force density levels L Ftrain dB( , ) for a train from Eq. (3), line-source transfer mobility levels TM L dB( , R, ) should be measured or estimated in advance. In this study, the point force amplitudes L F ( , ) of hammer impacting are unavailable and estimated based on the optimization of the objective function in Eq. (4). The identification of line-source transfer mobility is achieved by superimposing point-source transfer mobilities of each hammer impact based on the estimated point force amplitudes. Figure 6a and b show the line-source transfer mobility levels from 7 hammer impacts at the railhead to the ground surface receiver locations Loc 1 and Loc 2, respectively. The estimated averages were made based on train-induced ground vibrations from 10 different train passages on the hammer-impacted track. The 95% confidence intervals were displayed in grey shadow. The line-source transfer mobility reflects the vibrational energy transmission efficiency within the measurement site which can be attributed to the ground dynamic properties. It is seen that line-source transfer mobilities at Loc 1 and Loc 2 have similar patterns because of the location vicinity. The ground-borne vibration transmission is favorable in the frequency range of 25 to 100 Hz, which peaks at 63 Hz. By comparing the average between Fig. 6a and b, it is found that the vibrational energy transmitted to farther distances was less, especially for higher frequency components.

Identification of force density levels for metro train
The force density levels can be evaluated with Eq. (8) based on measured train-induced ground vibrations and the estimated line-source transfer mobility levels as shown in Fig. 6.  and 7, the main vibration energy at 50 Hz at Loc 1 was attributed to the train excitations related to the train and track dynamics, while the main energy at 63 Hz resulted from the peak path transmission efficiency related to the ground dynamics.

Numerical validations of force density levels
Wheel-rail contact forces are essential to the generation of train-induced ground-borne vibrations. Previous research (Lombaert and Degrande 2009) has discovered that traininduced ground vibration is mainly dominated by the dynamic axle loads compared to quasi-static loads for the case where the train speed is much slower than the soil Rayleigh wave velocity. The train speed is approximate 6 m/s according to Fig. 4, which shows the passing duration of a car is around 3 s. The Rayleigh wave speed in the top soil layer at the measurement site is around 176.8 m/s based on shear wave speed and Poisson's ratio listed in Table 1. Thus, only vertical dynamic axle loads due to random rail unevenness are analyzed in this section. In the following, the relationship between the force density and dynamic axle load was described analytically at first, to give an insight into how the force density reflects the train-induced vibration generation mechanism. The dynamic axle load caused by the unevenness at the wheel-rail contact surface was subsequently predicted with a train-track vertically coupled dynamic model, through which the fix-point loads assumption in the theoretical derivations was justified.
Assuming the stationary part of train-induced ground vibrations was equivalently caused by dynamic axle loads applied at fixed positions on the rail (Verbraken et al. 2011), the root mean square velocity levels in one-third octave band can be expressed as: where S F ( ) is the mean power spectral density for all axles. The upper and lower limits in the integration are determined by the one-third octave band center frequency and its corresponding bandwidth. zz ( , R, ) is the complex impulse response function matrix. Each entry of it represents the vertical velocity response at the receiver location R due to a vertical impulse exerted at the corresponding location in loading position set . L Vref dB is the reference velocity in decibel, which equals − 160 dB. According to the FTA guideline (Quagliata et al. 2018), the influences of vibration excitations and transmissions on train-induced ground-borne vibrations are independent of each other, which leads to the relationship between force density levels and PSD of dynamic axle load can be expressed in Eq. (10) (Zhai et al. 2009).
Fixed-point loads assumption was made for identifying train-induced vibration generation and transmission characteristics from field measurements, which have been schematically drawn in Fig. 8. The moving wheelrail contact forces have been approximated by equidistant fixed-point loads within the length of a vehicle or a whole train. To justify this key assumption behind the identification methodology proposed in the "Identification method for force density and line transfer mobility" section, the dynamic axle loads were numerically predicted using the train-track coupled dynamic model, whose power spectral density was then calculated and transformed with Eq. (10) to make a comparison between numerically simulated and data-driven identified force density levels.
The train-track coupled model has been maturely used (Wang 1988) to understand the dynamic axle load caused by the rail unevenness. The vehicle is modeled with the 10-DOFs multi-body sub-model as graphed in Fig. 8. The degree of freedom in the vehicle sub-model contains vertical translations of the car body, two bogies, four wheels, and the rotations of the car body and two bogies. The system equation is displayed in Eq. (11).
where c , c , c , c , c are the system mass matrix, stiffness matrix, damping matrix, external force vector, and system unknown displacement vector, respectively. The detailed expressions for the vehicle subsystem matrices can be found in Zhai et al. (2009).  The track was modeled with Euler's beam discretely supported by two layers representing sleepers and the ballast, respectively. The track subsystem contained FEM-based assembled elements, each of which has 6 degrees of freedom including vertical translation and rotation at the left and right ends of the rail segment, and vertical translations at the sleeper and ballast mass. The dynamic wheel-rail contact forces were the bridge to couple vehicle and track subsystems that follow the nonlinear Hertz contact theory and account for the rail unevenness external excitation, which can be calculated with Eq. (12).
where G = 3.86R −0.115 × 10 −8 m∕N 2∕3 is the parameter related to the wheel tread profile of the metro train. R is the radius of the wheel in meters. z wi , z ri , z ui represent the wheel displacement, rail displacement, and rail unevenness at i th wheel-rail contact location, respectively. (12) The dynamic properties of the studied metro train and ballast track are listed in Tables 2 and 3, respectively. A total of 200 track elements were used to simulate a track length of 109 m.
The rail unevenness (Tao et al. 2023) sample used was constructed based on the PSD of the short-wavelength rail surface irregularity (Zhai et al. 2009;Wang 1988) that expressed in Eq. (13): where S(f ) is the power spectral density of the rail unevenness in unit of mm 2 /(1/m), and f is the spatial frequency in unit of 1/m. Considering the vehicle speed (6 m/s) and frequency range of interest ([12.5 Hz,100 Hz]), the lower and upper spatial frequencies were set to be 1 and 30 (1/m) corresponding to 6 and 180 Hz, respectively. The variation of vertical rail unevenness along the track in the direction of train operating is shown in Fig. 9a. Using Eq. (10), the numerically predicted time-dependent wheel-rail contact forces that are shown in Fig. 9b can be transformed to a force density level scale. Figure 10 compared the numerically simulated and the data-driven identified force density levels. The simulated force density levels were overall comparable to the identified force density levels. The difference in peak force density level between simulation and identification was only 1.2 dB. It justified the fixed-point loads' assumption in FTA-based train-induced vibration prediction formulation, as well as the feasibility of the proposed method in identifying the force density levels for the train.

Experimental validations of predicted train-induced vibrations
The second step of forward prediction of train-induced ground and structural vibrations was carried out in this section, based on the previously identified line-source transfer mobility levels TM L dB( , R, ) and force density levels L Ftrain dB( , ) shown in Figs. 6 and 7. Ground vibrations were predicted at 3 m and 9 m away from the track, where the predicted locations were different from the two locations for identification, i.e., Loc 1 and Loc 2 shown in Fig. 1. The proposed method based on Eq. (1) is used for predicting train-induced vibrations at the ground surface, which cannot be used directly to predict structural vibrations since there is soil-structure dynamic interaction loss (Tao et al. 2023). However, the predicted vibrations at the ground surface can be adopted further as inputs to the building when adjustments regarding the soil-structure interactions (SSI) are accounted for. A 10 dB coupling loss for the building with piled foundations was selected based on the FTA guideline to predict train-induced structural vibrations. Train-induced vibrations at platform column bases located 3 m and 9 m away from the track were also  predicted. Predictions and measurements are compared in Figs. 11 and 12 for locations at 3 m and 9 m away from the track, respectively. A total of 15 validation train passages that were different from the 10 identification passages were adopted for the comparison.
The spectra shown in Figs. 11 and 12 by thin dotted lines were for individual 15 validation train passages of the same type of metro train operated on the same track. The black solid line is the corresponding average of the validation measurements.The grey shadow area represents the 95% confidence interval statistically derived from the measurements. The variability displayed in the ground vibrations induced by different train passbys was related to the randomness from both the vibration excitation and transmission. The red solid line is the corresponding predicted spectrum of train-induced velocity level using the averaged line-source transfer mobility levels TM L dB( , Loc1, ) and force density levels L Ftrain dB( , ) . It was seen in Figs. 11 and 12 that the predicted vibrations captured the measured tendency at both ground and column locations and at different distances away from the track. Although there were discrepancies resulting from the soil inhomogeneities and vibration uncertainties, the predicted results were primarily located within the 95% confidence interval of measurements. The predicted vibrations at the column bases displayed larger differences compared to that at the ground surface because the frequency-independent coupling loss of 10 dB was used. If a more detailed and sophisticated SSI analysis (Tao et al. 2023) is considered, the prediction accuracy for train-induced structural vibrations would be improved.

Conclusions
A hybrid method comprising field measurements and the empirical formula is proposed in this research for predicting train-induced vibrations. The proposed method follows a twostep methodology combining back-estimation and forward prediction. The method is practical and efficient, especially suitable in the vibration screening phase since it is formulated in the frequency domain and the time-consuming modeling work is avoided. Instead, the line-source transfer mobility and force density are identified based on acceleration measurements at the ground surface and the empirical formula. The identified transfer mobility and force density characterize the vibration transmission and excitations, respectively, where uncertainties are embedded intrinsically because of the random nature of field measurements. The back-estimated transfer mobility and force density are utilized straightforwardly in the second step for predicting train-induced ground and structural vibrations.
A case study was carried out at Shenzhen Metro in China. The identified transfer mobility and force density showed that the frequency components of 50 and 63 Hz of ground vibrations at 3 m away from the track were primarily attributed to train excitations and soil transmission efficiency, respectively. Comparisons of numerically predicted and identified force density levels were conducted for validating the first step of the proposed method, i.e., the back-estimation step. In addition, predicted and measured train-induced vibrations at the ground surface and structures were compared to validate the second step of the proposed method, i.e., the forward-prediction step. Both numerical and experimental comparisons show good accuracy that validated the feasibility of the proposed method. The identified force density of the metro train in the case study can therefore be referred to by similar railway systems.