Building Vibration Prediction Induced by Moving Train with Random Forest

)is paper adopts a combination of numerical simulation, field test, and Random Forest to predict the building vibration induced by moving train. First, a three-dimensional finite element model based on train-track-site soil-building system is established, and the track dynamic reaction force calculated by the train-track model is applied as an excitation to the site. On the soil-building model, this paper analyzes the influence of train speed, axle load, site soil characteristics, and distance from the building on the vibration of the building caused by the train. With the Random Forest, these different influencing factors are used as inputs, and the building vibration is the output. )us, the prediction model of the building vibration caused by moving train is established. )e prediction accuracy can be tested with the measured data. )e results show that this prediction method can provide a higher prediction accuracy with the maximum error (less than 6.41%) and the average error (less than 2.29%). )is method overcomes the shortcomings of traditional prediction methods and improves the accuracy of vibration prediction.


Introduction
Rail transit has developed rapidly in China in recent years due to its advantages of large capacity, high speed, and low transportation cost. High-speed rail transit also causes more and more serious environmental vibration problems. Vibration damage has become one of the seven major public hazards [1]. Environmental vibration will not only cause serious insomnia, anxiety, and other negative effects on individuals but also affect the accuracy of precision instruments. e research on environmental vibration caused by rail transit has attracted many scholars' attention [2][3][4][5]. e building vibration induced by moving train has also been studied by some literature. Kouroussis et al. [6] used finite element modeling for evaluating the ground waves propagation and a coupled lumped mass model of the soil to avoid computational resources. And then they proposed modal analyses of the vehicle and the building to acquire the effects of seismic wave amplification. Coulier et al. [7] tried to predict the vibrations in buildings by railway traffic. In their study, a coupled finite element-boundary element methodology is used to analyze the interaction between a building and a railway tunnel. ey found that the interaction between a source at depth and a receiver mainly affects the power flow distribution. is mainly due to the fact that the distance between source and receiver is smaller than the dilatational wavelength in the soil. Lopez-Mendoza et al. [8] proposed a scoping model to predict building vibration induced by railway traffic. e input of the model is the data of ground-borne vibration as model input. Galvin et al. [9] also adopted scoping assessment to predict freefield vibrations by railway traffic. Ding et al. [10] aimed to study the soil vibration response of a pile group to reflect the group-soil vibration induced by railway traffic. Paneiro et al. [11] measured ground vibrations caused by urban rail transit, trained on the measured data through artificial neural networks, and used artificial neural networks to predict vibrations. However, the influence of soil characteristics on vibration propagation is not considered in this paper. Connolly et al. [12,13] proposed a vibration prediction method.
Recently, some researchers [14][15][16] used swarm intelligence to predict the vibration induced by railway traffic. Jayawardana et al. [17] used artificial neural network (ANN) to predict the vibration isolation effect of double-layer vibration isolation trenches and developed a general prediction model. Yao et al. [18,19] used the support vector machine to predict the vibration of the building through the measured data and the finite element analysis method. Fang et al. [20] used numerical simulation results and field test data to train artificial neural networks to improve the accuracy of vibration prediction. eir prediction model studied the vibration of the site soil but did not analyze the vibration of the building.
is paper tries to propose a vibration prediction model based on Random Forest. en the numerical simulation method and the field test method are used to improve the prediction performance. e remainder of this paper is organized as follows. Section 2 describes our numerical analysis model which consists of a train-track model, a site soil model, and a building model. Building vibration influence factors are discussed in Section 3, followed by a description of the prediction model based on Random Forest in Section 4. Finally, Section 5 concludes this paper.

Numerical Analysis Model
A three-dimensional model of the train-track-site soilbuilding system was established to shorten the calculation time.
e whole system is divided into three subsystem models: train-track, site soil, and building models. Using the train-track system model, the vertical dynamic reaction force of the track on the site soil can be obtained when the train is moving. e vertical dynamic reaction force is loaded on the site soil model, and the vibration response of any point in the site can be obtained through calculation. e vibration of the building in the site soil model to the foundation of the building can be obtained based on the vibration from each point of the building.

Train-Track Model.
e train is simulated as a multirigid and multidegree-of-freedom system with moving on the track at a certain speed. Since the impact of vertical vibration on the surrounding environment is greater than that of lateral vibration, this paper only considers the impact of vertical vibration on buildings. A train carriage rigidly connected to the rail using Euler beam model simulation, discrete point support manner, the base rail using ballasted track model, and the Hertz nonlinear elastic contact are used to analyze the coupling between the train and the track [21].

Train Model.
e train model is composed of multiple vehicles, and each vehicle is considered to be composed of multiple rigid bodies with concentrated mass, including carriages, bogies, wheelsets, springs, and dampers. e train calculation model is shown in Figure 1. e parameters of each component are shown in Table 1. e train and the bogie, the bogie, and the wheels are all connected by springs and dampers to simulate the suspension system of the train. In the model, the pitch (βc) and float (Zc) of the train, the pitch (Zt1, Zt2) and float (βt1, βt2) of the bogie, the vertical movement of the four wheels (Z wi , i � 1∼4), a total of 10 degrees of freedom, and Lagrangian equation are used to establish the train's motion equation, as shown in formula (1): In the formula, T, V, and Q, respectively, represent the total kinetic energy, total elastic potential energy, and total damping dissipation energy of the motion system.
Based on the Lagrangian equation, the equation of train body, bogie, and wheelset can be established. e motion equation of the i-th carriage and its corresponding two bogies is shown in equation (2): In formula (2), Z i is the displacement vector of the i-th train; P i (t) is the external excitation force of the i-th train; M i , C i , and K i , respectively, represent the mass of the i-th carriage, damping, and density.
In the train-track model, the random vibration of the train is simulated by inputting the track irregularity spectrum. e track irregularity spectrum is based on the low interference power density function of German high-speed line track irregularity [22], which is obtained by inverse Fourier transform with MATLAB software.

Track Model.
is paper uses the form of ballasted track. e track system consists of four parts: rails, sleepers, track beds, and subgrades, which are simulated as a threelayer discrete model. e track is regarded as an Euler beam model supported by discrete points, which are discrete according to the support points of the sleepers. Each element is composed of two mass units (M si and M bi ) and three-layer spring damping units (K pi and C pi , K bi and C bi , and K fi and C fi ). Adjacent units are connected by shear spring damping (K wi and C wi ). e model is shown in Figure 2.

Site Soil Model.
e finite element analysis software ANSYS was used to establish the soil calculation model of the site. In order to shorten the calculation time, the site soil model was simplified. e line length was 48.5 m, vertical track width was 120 m, and vertical track depth was 52.9 m.
ere is a two-layer direction perpendicular to the track; the size of the model is shown in Figure 3.
In the actual working conditions of the site, the site soil is a multilayer structure, and the characteristics of each layer of soil are different. In the numerical simulation, in order to reduce the calculation conditions and shorten the calculation time, using formula (3), the weighted average method is used to convert multiple layers of soil into two layers of soil [23] for simplified calculation. e converted site soil parameters are shown in Table 2.
e right boundary and the lower boundary in the model adopt a three-dimensional uniform viscoelastic artificial boundary to reduce the impact of vibration wave reflection at the boundary on the vibration response [24]. e front boundary and the rear boundary are regarded as free boundaries.
e left boundary adopts a symmetrical boundary.

Building Model.
e building is a frame structure type with 6 stories, each of which is three meters high. e total height is 18 m, the length is 3 * 12 � 36 m, and the width is 1.4 m. e span is 6 m; the intermediate span is 2 m. e building consists of three parts: beam, column, and plate. e cross-sectional size of the column is 500 mm × 500 mm; the cross-sectional size of the main beam of the house frame is 350 mm × 600 mm; the cross-sectional size of the secondary beam is 350 mm × 500 mm; the plate thickness is 10 cm; the building layout is shown in Figure 4. e frame beams and columns of the building are made of C25, and the floor slab is made of C20 concrete. e building damping ratio ξ c � 0.05. e finite element model of the building is shown in Figure 5. e large mass method is used to load the building, and the spring elastic connection is used between the concentrated mass and the base node of the building without considering the damping effect. e spring stiffness is calculated according to formulae (4) and (5): where Z is the distance from the contact surface to the free surface. In this paper, the building is placed on the site soil surface, so the value Z is 0. λ and μ are lame coefficient. k α , k β , and k c are P wave, S V wave, and R wave number, respectively. T and R are wave transmission coefficient. e letters with "′" superscript are all the parameter characteristics of building foundation, and the definition of parameters is the same as that of soil. ese parameters can be obtained by experiment before calculation. e calculation results K X and K z obtained by formulae (4) and (5) are spring stiffness corresponding to unit soil area. e spring stiffness in the model is determined by the average area of the four soil elements around the spring element and the spring stiffness coefficient K X or K z . e  stiffness of each spring element in the model is the product of the spring stiffness coefficient and the average area of the four elements around the node. In this paper, the x-axis and y-axis of the coordinate system are perpendicular to each other, the x-axis and y-axis are parallel to the Earth's surface, and the z-axis is perpendicular to the Earth's surface. erefore, the spring stiffness coefficient in the y-axis direction is the same as that in the x-axis direction.
e train-track model calculates the dynamic reaction force of the track. And then the reaction is put on the finite element model of the site soil. e displacement, velocity, and acceleration of any point on the site soil can be obtained. e building model is loaded by the mass method, and the mass is taken as the structural quality 1E6 times [25]. e product of the vibration acceleration and the concentrated mass is the force between the soil and the building. By analyzing the finite element model of the building, the vibration response of the building can be obtained.
e site soil and building are modeled independently for calculation and analysis. rough the calculation of the site model, the vibration response of each node in the site model can be obtained. When the building is loaded, the vibration acceleration of the node corresponding to the center of the building in the site model is extracted, and then the extracted vibration acceleration is loaded to the node at the bottom of all the columns of the building according to the large mass method.

Analysis on Building Vibration Influence Factors
In order to analyze the influence of train speed, axle load, distance, soil damping ratio, density, elastic modulus, Poisson's ratio, and so on on the building vibration caused by the moving train, these parameters need to be analyzed. e analysis adopts the controlled variable method and only adjusts the vital parameters like vehicle speed, train axle load, distance from buildings, and site soil characteristics. Other parameters remain unchanged, the middle point of each layer is selected as the calculation point, and the vibration level (dB) is used as the evaluation index. Various parameters are shown in Table 3.

Influence of Vehicle Speed.
Select the train speed as 67 km/h, 100 km/h, 134 km/h, 167 km/h, and 252 km/h. Figure 6 shows the curve of the acceleration vibration level of the building floor under different train speeds.
It can be seen from Figure 6 that when other conditions remain unchanged and when the train speed is 167 km/h, the vibration of the building is the smallest. When the train speed is 252 km/h, the vibration of the building is the largest. When the height increases, the change of vibration is not obvious, and the vibration of the building generally shows a trend of increasing with the increase of speed.

e Impact of Train Axle Load.
Axle load is selected for the train. Five cases as 122 kN, 128 kN, 144 kN, 160 kN, and 176 kN are selected for analysis. Figure 7 shows the vibration acceleration level of the building floor with different train axle loads.
It can be seen from Figure 7 that, with the increase of the train axle load, the vibration level of the same measuring point on the building floor gradually increases. e train axle load has a certain influence on the building vibration. When the train axle load increases from 122 kN to 176 kN, the vibration level changes about 3 dB. It can be seen from Figure 8 that distance has a significant effect on building vibration. Under the same distance, the vibration level between different floors is not much different, but as the distance increases, the building vibration level caused by the train gradually decreases, and the decrease in vibration level is very obvious.

Influence of Soil Damping Ratio.
e soil with damping ratios of 0.05, 0.1, 0.15, 0.2, and 0.25 is selected for calculation and analysis. e train speed is 67 km/h, axle load is 160 kN, and the other conditions remain unchanged. Figure 9 shows the variation curve of the acceleration vibration level of the building floor under different site soil damping ratio conditions. It can be seen from Figure 9 that, with the increase of the site soil damping ratio, the floor vibration level gradually decreases, and the vibration level difference of the same calculation point is about 7 dB; as the damping ratio increases (from 0.05 to 0.10), the vibration level of the same point is different. e maximum vibration level of the same computing points also changes with the increase in the damping ratio (from 0.20 to 0.25).

Influence of Soil Elastic Modulus.
e multiples of the elastic modulus of the two layers are the same, and the multiples are multiplied by the elastic modulus values in Table 3, respectively, which are 0.1 times, 1 time, 10 times, 20 times, and 30 times. Figure 10 shows the fold change of the ground soil different elastic moduli and the vibration acceleration level of the building floor with floor layers' curve.
It can be concluded from Figure 10 that the vibration of the building does not increase monotonously with the increase of the soil elastic modulus. When the soil elastic modulus is 1 time, the vibration level of each floor calculation point is the largest. When the soil elastic modulus is 30 times, each floor vibration level is the minimum.

Influence of Soil Density.
Due to the large difference in soil layers, it is difficult to analyze with the same numerical value. e density of soil changes by the same multiple, and the product of multiples is taken as the standard of the soil density in Table 3. Take 0.75 times, 1 time, and 1.25 times, 1.5 times, and 1.75 times. Figure 11 shows the curve of the acceleration vibration level of the building floor with the floor under different density changes of the site soil.
It can be seen from Figure 11 that, with the increase of soil density, the vibration level of the same calculation point gradually decreases; for the same soil density change, the vibration level of the same calculation point changes greatly when the soil density is small.

Test Analysis.
Because numerical simulation needs to make idealized assumptions on the model to achieve the purpose of simplifying the model, there is an error between the numerical simulation results and the real vibration response. e field test data is the real vibration response, and the data results incorporate various influencing factors. Based on the Random Forest, the prediction model is deeply optimized using field test data to improve the accuracy of prediction.
e test site is located in a residential building near the Beijing-Guangzhou line. Passenger trains and freight trains are running on the line at the same time (this paper only analyzes the vibration response caused by passenger trains), the axle load of passenger trains is 160 kN, and the speed is between 40 and 115 km/h. e residential building at the test site is a 6-storey brick-concrete structure, with a total height of 17.25 m, a length of 58 m in the direction perpendicular to the line, and a width of 12 m in the direction parallel to the line. e distance between the building and the side rail of the line is 14 m. Each building has 4 units. e distance to the line is 4, 3, 2, and 1 units in order from far to near, the distance between each unit is 14.5 m, there is no basement, the storey height is 2.8 m, and each floor is made of 120 mm thick C25 cast-in-place concrete; the site layout is shown in Figure 12. e soil parameters are obtained by drilling and sampling in the laboratory. e soil quality parameters of the site where the building is located are shown in Table 4.
e measuring points are arranged on each floor of unit 4 of the building, as shown in Figure 13. Each measuring e test collected 25 sets of data that can be used in this paper. After the collected vibration acceleration data is filtered and smoothed, Fourier transform is performed on the collected data, and the transformed data is analyzed by 1/3 octave frequency, and then it is processed and converted into a vibration acceleration level, and the processed data is used in the training of the Random Forest program.

Data Processing.
e data obtained from the field test and numerical simulation is the vibration acceleration, and the data is processed and converted into a form with the vibration level as the evaluation standard. e calculation of vibration level is shown in formula (6): In formula (6), L a is the vibration level (dB); a ref is the reference vibration acceleration, which takes the value 1 × 10 −6 ; a (m/s 2 ) is the root mean square vibration acceleration (RMS); the calculation equation (7) is as follows:

Journal of Advanced Transportation
In formula (7) in, a frms is the frequency f vibration acceleration (RMS), in units of m/s 2 ; c f is corresponding to a different frequency vibration acceleration correction value.

Algorithm Development.
e Random Forest was first proposed by Breiman [26] and others. rough the learning of existing data, the regression prediction of unknown data is realized, and the accuracy of prediction is improved through the integrated method of decision trees. In this paper, regression tree is used as the base regressor for integration. Each decision tree is independent of each other and can be calculated in parallel.

Building a Decision Tree.
After the bagging method is used in the overall training set to collect a subset of training samples for each decision tree, each decision tree is trained to form a Random Forest. Each decision tree is trained independently, and no pruning is performed. e CART [27] algorithm is selected as the decision tree node splitting method, and the C ART algorithm uses the minimum mean square error sum as the calculation index for node splitting. e calculated attributes of each node are selected at random instead of all attributes, so as to effectively reduce the generalization error. e specific calculation method of CART is as follows: for any partition characteristic A, the corresponding arbitrary partition point s is divided into data sets D 1 and D 2 on   min min In formula (8), min x i ∈D 1 (A,s) (y i − c 1 ) 2 is the D 1 sample output value data set; min x i ∈D 2 (A,s) (y i − c 2 ) 2 is the D 2 sample mean output data set. e prediction of the C ART tree outputs the prediction result based on the average value of the leaf nodes, so the prediction of the Random Forest is the average of the predicted values of all trees.
In this paper, the method of randomly combining input variables is selected as the method of random feature selection, and multiple input variables are selected to determine the random feature by linear combination. At the node, each input variable is linearly added with the coefficient as the weight to generate n linear combinations, and then the CART algorithm is selected to split.

Algorithm Formation.
e Random Forest model is written by MATLAB. e process of building the model is as follows:   (1) Random sampling is performed from the training samples using the bootstrapping method. e sampling takes the form of replacement. Five samples are taken each time, and a total of 1736 samples are taken. Generate 1736 training sets.
(2) Train the corresponding decision tree models on 1736 data sets, respectively, and train 1736 times in total. (3) In each decision tree model, the number of features of the training sample is randomly selected. Assuming that the number of features is n, then the CART algorithm is used to select the optimal feature for each split and then split. (4) Each tree is split according to the above method until all the training samples on the node are of the same type. (5) Combine each generated decision tree model to form a Random Forest. e predicted value is calculated through each tree, and then the predicted value of each tree is averaged to determine the final prediction result. Figure 14 shows the procedure of the training of the proposed RF model.   (6) After changing the number of random forests, repeat the above 1∼5 steps and select the best number of random forests as 1320.

Parameter Selection.
In order to improve the calculation efficiency and shorten the calculation time, only the two soil characteristics that have the greatest impact on vibration are considered in the algorithm. In the sensitivity analysis of soil characteristic parameters, all parameters in the finite element model remain unchanged except for the characteristic parameters of the two-layer site soil (excluding the roadbed) in the site model. e single variable method is used to analyze the sensitivity of each parameter. e damping ratio, Poisson's ratio, elastic modulus, and density parameter values of the site soil were increased by 15%, and other parameters remained unchanged. Load the train wheel-rail force into the three-dimensional finite element model for calculation and analysis, and the acceleration time history data can be obtained and converted into vibration level. Compare the vibration level value after each parameter change with the result before the parameter unchanged, and then the difference is recorded. e parameter is used to represent the sensitivity. e sensitivity analysis of different parameters of the site soil is shown in Figure 15.
It can be seen from Figure 15 that, among all the parameters of the site soil, the soil damping ratio has the greatest impact on the building vibration, followed by the density. erefore, the damping ratio and density are chosen to represent the characteristics of the soil.

Instance Verification.
Select site soil vibration for prediction. Choose one of the working conditions to compare the test results, numerical model calculation results, and Random Forest prediction results. e working condition is that the train speed is 67 km/h, the train axle load is 160 kN, the upper soil elastic modulus is 17.7 MPa, and the lower soil elastic modulus is 16.9 MPa. e comparison result is shown in Figure 16.
According to Figure 16, we can get the following: (1) e difference between the predicted value obtained by the Random Forest method and the simulated value obtained by the numerical simulation method is small, the maximum error is 2.35%, and the average error is 0.72%. (2) e maximum error between the predicted value of the Random Forest method and the experimental data is 6.41%, and the average error is 2.29%, which is 1.19% and 0.58% lower than the error between the numerical simulation value and the experimental value, respectively. e results show that the Random Forest model can obtain data with higher accuracy than numerical simulation and can objectively reflect the law of propagation in the soil. training sample is relatively small, the predicted value of Random Forest is closer to the calculated value of numerical simulation.

Conclusions
is paper uses a combination of numerical simulation, field test, and Random Forest to establish a prediction model of building vibration caused by trains. e prediction model is used to predict the environmental vibration under a certain working condition and compare it with the test results. e following can be obtained in conclusion: (1) Numerical calculation results show that factors such as train speed, train axle load, distance between track and building, soil density, and soil damping have a greater impact on the environmental vibration response caused by high-speed trains. (2) Simplify the multilayer soil structure of the site into two layers for modeling analysis, which can not only consider the influence of multilayer soil but also simplify the model. (3) e method of combining the Random Forest with the three-dimensional finite element model and field test data can effectively make up for the shortcomings of the ideal assumption of the finite element model and insufficient field test data and improve the accuracy of prediction.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.