Modelling Ambient Vibration Responses Induced by Operation of Metro Train on Curved Rail Segment with Small Curvature Radius

The torsional effect of the tracks on curved segment intensifies the ambient vibration response induced by Metro operation. This paper studies the ambient vibration responses induced by the operation of the Metro train on curved rail segments. By taking the curved segment of Hangzhou Metro Line 1 of China as an example, the wheel-rail model employing the multibody dynamics is demonstrated; the dynamic wheel-rail force of a B-type vehicle passing through a 350 m radius curved segment is also calculated. A finite element method model of the track-tunnel-soil-building is developed and verified by comparing the measured results with the simulated ones. Then, we analyzed the ambient vibration responses induced by the Metro operation. The results show that the horizontal vibration induced by the Metro train, on curved segments, cannot be ignored. When propagating on the ground, the variation trend of the horizontal vibration acceleration is greater than that of the vertical vibration. The horizontal vibration attenuates faster than the vertical vibration. The secondary vibration along the horizontal direction yields a significant amplifying effect upon the building. The vibration level increases with the height of the building, along the horizontal direction, but the vertical vibration level changes negligibly. The insight exhibits the influence level of the horizontal action of the Metro train wheels to the tracks, which can be referred by the practitioners for the planning and operation.


Introduction
For the past several decades, urban rail transit has boomed in developing countries, such as China [1]. Metro can greatly relieve the pressure of the urban traffic; however, the Metro train-induced structural vibration, along with the ambient noise, can also cause adverse effects on humans, including physical, physiological, and psychological effects [2][3][4][5][6][7]. erefore, ambient vibration along Metro lines has already begun to gain increasing attention.
To analyze the ambient vibration response, obtaining the train vibration load is one of the most important steps; many scholars have done a variety of studies in this area by developing theoretical models or field test methods. As is known, Timoshenko [8] solved the classical problem of the rail dynamic stress based on the elastic foundation beam model, which is the basis of the research in this area. Auersch [9] established the vehicle-track interaction model and obtained the wheel-rail excitation force of the train. Field measurement and numerical simulation are both needed to analyze these specific engineering problems. Shao et al. [10] carried out numerical simulation by employing the multibody dynamics, compared with physical experiments to verify the effectiveness of their approach. Sheng et al. [11] put forward the coupling analysis method of the finite element and boundary element and obtained the dynamic response solution of the train vibration. Wolf [12] used the finite difference method software, FLAC 3D , to obtain the ground displacement variations under the Metro load and compared them with the measured values. Hussein et al. [13] exhibited the PiP model for attaining the vibration response from railway tunnel in a multilayered half-space.
In addition, a good number of the former studies focused on the propagation and attenuation laws of the ground vibration induced by the Metro operations. Sheng et al. [14] analyzed the attenuation law of the ground vibration intensity by employing a layered half-space model of the Metro. Yang and Hsu [15] found that the ground vibration caused by the Metro has an amplification zone in the attenuation process. Volberg [16] studied the propagation law and attenuation characteristics of the ground vibration and discovered that the surface wave plays a dominant role in the vehicle-induced vibration. Hayakawa et al. [17] studied the mechanism of the vibration induced by Metro train and the propagation law of the elastic wave on the ground and presented a prediction method of the surrounding environmental vibration caused by the Metro operation. Melke and Kraemer [18] found that when the vibration frequency of the Metro train in operation approaches the inherent frequency of the surrounding soils, the acceleration of the ground surface has a significant increase due to the resonance. e vibration response of the buildings, caused by the Metro train in operation, has also been broadly studied. Based on the impulse excitation and experimental analysis, Melke and Kramer [19] studied the vibration response of the adjacent buildings around Metro lines. Xia et al. [20] found that the vibration level of the building floors increased with the train speed and decreased with the distance away from the railway; moreover, the vibration in high-rise buildings is stronger than that of the low-rise buildings. Okumura and Kuno [21] studied the influence of the vibration caused by the high-speed train operation on the surrounding buildings, discovering the main influence factors which include the track centerline, train speed, and length of the train.
Note that more and more Metro lines with curved segments have been developed in big cities, especially in China. However, so far, limited studies have focused on the ambient vibration response caused by the trains running on curved segments. Former investigations suggest that the horizontal vibration of the train traveling on the curved segment might enhance the amplitude of the ambient response [22]; such case could be intensified when the curve radius is small, generally meaning less than 650 m [23]. However, to what extent the amplification effect can reach still needs to be further studied.
In this paper, taking a curved segment of Hangzhou Metro Line 1 of China as an example, we demonstrated a wheel-rail model by employing the multibody dynamics method, obtaining the dynamic wheel-rail force of the B-type vehicle passing through a curved segment with a radius of 350 m. Subsequently, the FEM model of the tracktunnel-soil is developed. After validating the model by comparing with the measured results, the ambient vibration response, including the building vibration response, induced by Metro trains on the curved segment with a small radius, is analyzed. e investigation may enhance our understanding of the vibration responses caused by the Metro operation on curved segments.

Mathematical Equations of the Wheel-Rail Creep Force.
e model analysis method is a method to calculate the wheel-rail creep force of the train by using the finite element software, simulation software, and so on. It is based on the wheel-rail coupling dynamics.
e system model mainly consists of vehicle system, track system, and wheel-rail model, as shown in Figure 1.
e vibration behavior of a Metro train on a curved segment is more complicated than that on a straight segment because of the side wear and wave wear [25,26].
When a train moves on a straight-line segment, the horizontal effect of the track on the train is usually neglected. e vertical wheel-rail force can be determined by Hertzian nonlinear elastic contact theory equations [27] as follows: where G is the contact rail constant and δ · Z is the wheelrail elastic compression. When a train runs on a curved segment, the wheels collide with the track surface, owing to the irregularity of the track and centrifugal force, which results in the wheel-rail contact force. e action point is at the wheel-rail contact, as shown in Figures 2(a) and 2(b).
As seen from Figure 2(b), the wheel-rail vertical force P can be regarded as the vertical component of the wheel-rail normal force K and creep force Q, while the wheel-rail horizontal contact force W can be regarded as the combination of the horizontal component of K and Q, as follows: where K(p) and K(w) are the vertical and horizontal component of the wheel-rail normal force K, respectively, and Q(p) and Q(w) are the vertical and horizontal component of the creep force Q, respectively. e wheel-rail creep force Q is related to the vertical, horizontal, and torsional accelerations of the left and right wheel-rail pairs. e corresponding formula yields where L(v) and R(v) are the vibration speed related to the left and right rail.

Vibration Level Formula.
Herein, we use the vibration level to describe the variation range of the vibration magnitude of the Metro vibration. e formula for the vibration acceleration level [28] yields where a 0 is the reference acceleration. Typically, a 0 � 10 − 6 m/s 2 .
a w is the acceleration effective value, also known as the acceleration root mean square value. e formula yields     Figure 1: Train-track dynamic analysis model [24].    where a w (t) is the acceleration time history and T is the measurement duration.  13.1% of the total distance. Among them, the curved segment of Yan'an Road between Longxiangqiao Station and Anding Road Station passes through residential areas. e target building near the curve segment is 30 m away from the centerline of the right tunnel, 10 m wide, and 3 m high for each floor. As shown in Figure 3, the plane radius of the line is 350 m, the inner diameter of the Metro tunnel 5.5 m, the distance between the two tunnels 12.5 m, and the speed of the Metro 60 km/h. e elastic short sleeper integral ballast bed is adopted. e soil layers mainly consist of the filling soil, silty clay, and silty sand.

FEM Model.
e two-dimensional FEM model of the track-tunnel-soil-building is developed by PLAXIS software.
e theory of the dynamic module analysis in PLAXIS software is based on linear elastic model, so all constitutive models in the software can be used for dynamic analysis. e model is located in perpendicular to the track and tunnel. e horizontal distance of the model is 100 m and the height is 60 m. For efficiency, the complex soil layers are simplified to five soil layers, including the rockfilling, silty clay, muddy clay, silty clay, and silty soil mixed with silty sand. e Mohr-Coulomb constitutive model is adopted, and the damping ratio is set as 0.03. On the left and right boundaries of the model, the horizontal direction does not allow the displacement occurrence, whereas the vertical direction permits the deformation occurrence. To avoid the reflection of the elastic waves, viscous absorption boundaries are set on the left, right, and lower sides of the model. A neighbor four-floored building is additionally developed to analyze the ambient vibration response, as shown in Figure 4. e building is 30 m away from the centerline of the right tunnel, 10 m wide, and 3 m high for each floor. e building is a frame structure with isotropic plate elements for walls and floor. e foundation of the building is independent foundation, which is simplified as plate element in the model. Rayleigh damping is used for material damping, in which Rayleigh α and Rayleigh β are 0.128 and 0.007, respectively. e strength grade of underground rail concrete is C30; the strength grade of ground rail concrete is C40. Double-layer reinforcements and longitudinal reinforcements adopt φ14HRB335 threaded reinforcements. e stirrup uses φ10HPB235 plain round reinforcement, and φ10HPB235 frame reinforcements are set between the two layers of the reinforcements. e rail adopts DTVI2-1 type fastener. e sleeper dimensions are 220 mm in width, 160 mm in thickness, and 2500 mm in length. e sleepers have larger resistance and rigidity in the transverse and longitudinal directions, smoother disturbance at the rail bottom, and smaller dynamic gradient. e sleepers have high elastic       cushions, which can ensure the uniform elasticity and durable operation life of the track. In order to get more accurate results, the grid density at the important structure is increased. e model consists of 937 units and 8,165 nodes, as shown in Figure 5. e building material parameters are listed in Table 1. e soil layers and tunnel parameters are listed in Table 2. And the fastener parameters are listed in Table 3.      [29][30][31][32]. e primary methods to determine the dynamic load of the Metro are field measurements, numerical simulation, and empirical analysis. In this paper, a numerical model of the wheel-rail loads is demonstrated by using the multibody dynamics software, SIMPACK, taking the B-type vehicles, commonly used in urban rail transit in   Table 4. Figure 6 depicts a simplified Metro train model. A metro train is mainly composed of the body, bogie, wheel, and suspension system; the body and bogie are simulated by rigid body, and the suspension system is simulated by spring damping element. Figure 7 is the track structure model. e track structure is a layered structure with alternate elastic layer and rigid layer, mainly composed of the rail, fastener, sleeper, and track bed. e rail is simulated by beam element, which has the function of supporting and guiding. e elastic element is used to simulate the fastener; its elastic modulus is mainly provided by rubber and has high internal resistance; the function of the fastener is to fix the rail on the sleeper, keep the track gauge, and prevent the rail from moving vertically and horizontally. e sleeper and sleeper plate are simulated by beam element, which has the function of making the pressure of the train spread evenly to the foundation.
e random irregularity samples of the track are obtained by using the five-level track irregularity spectrum of the United States, as shown in Figure 8. e dynamic wheelrail forces of a single trailer car and a single motor car with different load scenarios are obtained, as shown in Table 5. Train mechanical parameters are shown in Table 6.
Assuming that the dynamic wheel-rail forces of the vehicles with the same structure and different positions have only phase difference, the dynamic wheel-rail forces of all the motor vehicles and trailers are calculated according to the delay between the rear vehicle and the front vehicle (speed/workshop distance). For the conciseness, the scope of this paper only focuses on the wheelrail force of the first wheel pair of the overcrowdingscenario trailer, as shown in Figure 9. When the train dynamic load is applied to the FEM nodes of the tracktunnel-soil-building model as the load spectrum in the time domain, the vibration propagation in the underground structure can be calculated.

Model Verification.
In order to effectively reduce the environmental background noise caused by the flow of people and ensure the accuracy of the data, the field   Mathematical Problems in Engineering 9 measurements are arranged during the low passenger flow period of daytime subway operation, and each measuring point is tested many times. Four vibration receiver points, P 1 , P 2 , P 3 , and P 4 , are selected on the ground, located directly above the right tunnel in Figure 4. e points are 0 m, 10 m, 20 m, and 30 m, respectively, away from the center line of the right tunnel. In this paper, the numerical model is validated through field measurements. e monitoring system consists of the accelerometers, data acquisition system, gateway node, and notebook computer. Figure 10 illustrates the configuration of the monitoring system. Data acquisition was conducted using a JM3870 wireless dynamic/static vibration analysis system, with a sampling frequency of 256 Hz. e AI050 piezoelectric accelerometer is used, whose parameters are shown in Table 7. e test area starts directly above the center line of the right tunnel, and a receiver point is arranged every 10 m, with a total of four receiver points, as shown in Figure 11.
For the conciseness, only the measured and simulated acceleration curves in the both time and frequency domain of P 0 were listed. Figure 12 shows the measured and   simulated acceleration curves of P 0 in the time domain. Figure 13 shows the measured and simulated acceleration curves of P 0 in the frequency domain. Obviously, the general trend of the simulation and measurement is consistent. In Figure 12, the horizontal vibration acceleration is larger than that of the vertical vibration acceleration. In Figure 13, the measured acceleration frequency of P 0 mainly concentrates on 5-15 Hz, while the simulated acceleration frequency concentrates on 4-18 Hz, both in the low frequency band. e measured value is slightly larger than the simulated value, which may be effectuated by the external environmental impacts when the ground vibration is measured. In horizontal and vertical directions, the error between the simulation and measurement is slight, suggesting that the numerical model is reasonable. e ground response of the Metro train meeting with overcrowding scenario is analyzed; Figure 14 shows the contour plots of the horizontal acceleration and displacement, vertical acceleration, and displacement of the simulated overcrowding scenario. e displacement is related to the variation in the trend of the acceleration, so only the acceleration contour plot is presented here. e results show that the vibration acceleration generated by the Metro running on the curved segment has a certain amplification region in the soil layers. e acceleration does not decrease monotonously with the increase of the distance from the center line of the tunnel. e vertical acceleration tends to distribute symmetrically on both sides of the tunnel, and two amplification regions exist. Horizontal vibration acceleration has a smaller magnification area in the soil layer on both sides of the building, directly below the building. Horizontal acceleration attenuates faster than vertical acceleration. Note that the absorption of the horizontal elastic waves by the soil layer is stronger than that of the vertical ones.   induced by the Metro operation does not attenuate with the height of the building, and the horizontal secondary vibration has a significant amplification effect upon the building. Figure 19 shows the peak acceleration, whereas Figure 20 shows the vibration level for each floor under different scenarios. Obviously, the change rules of the vibration level are consistent with that of the peak acceleration; hence, we analyze the vibration level to exemplify the change rules.

Vibration Level of the Building.
As seen from Figure 20, the horizontal vibration level increases gradually with the height of the building, increasing by 2-5 dB for each floor, while the vertical vibration level does not change significantly with the height of the building. By the mutual comparison among the floor vibration levels from scenarios (left, right, and meeting) of the same train with different load scenarios (personnel quota and overcrowding), we found that the overcrowding scenario of the same floor is 1-2 dB greater than that of the fixed one in both horizontal and vertical directions. e vibration level of the building on the same floor is 0.5-2 dB greater than that on the left. Compared with the single-track

Conclusion
In this paper, we used the mathematical equations of the wheel-rail creep force and vibration level formula and developed the wheel-rail force model of a B-type car passing through the curved segment based on the multibody dynamics. Taking a curved segment of Hangzhou Metro Line 1 as an example, a tunnel-soil-ground-building model has been demonstrated, and the ambient vibration response induced by the Metro trains on a curved segment with a radius of 350 m was analyzed. e conclusions are as follows: (1) e external rail wheel-rail force is obviously higher than the internal rail wheel-rail force. e vertical force of the outer rail is generally 10-15 kN higher than the vertical force of the inner rail, and the horizontal wheel-rail force of the outer rail is 5-10 kN higher than the horizontal wheel-rail force of the inner rail. is is related to the centrifugal force produced by the train running in the curve segment. e horizontal wheel-rail force is close to half of the vertical force, so the horizontal effect cannot be ignored. (2) e variation trend of the horizontal vibration acceleration with distance is larger, while the vertical vibration acceleration is gentler, and the peak value of the vertical vibration acceleration appears rebound at 20 m. e horizontal vibration level above the tunnel is stronger than 90 dB. With the increase of the distance, the vibration level gradually decreases to about 75 dB at 30 m away from the tunnel. Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.