Vibration Isolation of a Rubber-Concrete Alternating Superposition In-Filled Trench for Train-Induced Environmental Vibration Based on 2.5D Indirect Boundary Element Method

A new train-induced vibration isolation measure of rubber-concrete alternating superposition in-filled trench is presented in this paper. For analyzing the vibration isolation effect of the new measure, this paper establishes a 2.5D train-track-layered foundation-filled trench model to analyze the dynamics of track and layered foundation with the in-filled trench. The correctness of the model is verified by using the measured data of the Sweden X-2000 high-speed train. The vibration isolation effect of the rubber-concrete alternating superposition in-filled trench is calculated by using the actual soft soil foundation parameters of the X-2000 high-speed train, and the vibration isolation effect is also compared with that of the empty trench, rubber in-filled trench, and concrete in-filled trench. The results show that the vibration isolation effect of the rubber-concrete alternating superposition in-filled trench proposed in this paper is better than that of the C30 concrete in-filled trench, especially the impact on displacement. Compared with low-frequency vibrations generated by the lower train speed, the rubber-concrete alternating superposition in-filled trench has a better vibration isolation effect on high-frequency vibrations caused by higher-speed trains. The rubber-concrete alternately superposition in-filled trench has the frequency band characteristics of elastic waves. Elastic waves in the passband frequency range can propagate without attenuation, while the elastic waves in the forbidden frequency range will be filtered out.


INTRODUCTION
With the large-scale construction of rail transit, the environmental vibration induced by train operation has attracted more and more attention. Research on vibration isolation of environmental vibration induced by train operation plays a guiding role in the design and construction of rail transit, and can effectively improve the quality of life of residents near the railway.
In the past, train vibration isolation was mainly by excavating a vibration isolation trench, and the trench was usually empty or in-filled with concrete. In 1968, Woods [1] was the first to summarize the basic criteria for barrier vibration isolation trench design through many experiments. Yang and Hung [2] conducted a parameter analysis on vibration isolation effects of empty trench and concrete in-filled trench. With [3] studied the isolation effect of vibration isolation measures on far-field vibration caused by trains. Since then, scholars have analyzed the vibration isolation efficiency of empty trench and concrete infilled trench in the uniform half-space foundation [4][5][6][7][8][9], the layered half-space foundation [10][11][12][13][14][15], the homogeneous saturated foundation [16][17][18][19][20][21][22], and the layered saturated soil foundation under moving loads.
The above studies are all aimed at empty trenches and concrete in-filled trench, which sometimes fail to achieve a good vibration isolation effect [23][24][25][26][27][28]. For this reason, this paper puts forward the vibration isolation measures of rubber-concrete alternating superposition in-filled trench. In this paper, a 2.5D train-tracklayered foundation-filled trench model is established to analyze the dynamics of track and layered foundation with the in-filled trench, and the correctness of the model is verified by using the measured data of Sweden X2000 high-speed train. Finally, the vibration isolation efficiency of rubber-concrete alternating superposition in-filled trench is calculated using the actual soft soil foundation parameters of Sweden X2000 high-speed train, and the vibration isolation efficiency is compared with that of the empty trench, rubber in-filled trench, and concrete in-filled trench.

MODEL AND METHODOLOGY
As shown in Figure 1A, the train tracks are laid on a layered foundation including a rubber-concrete alternating superposition infilled trench. The in-filled trench is W in width and H in depth and is parallel to the train track with the distance D. The train track is simulated as a Euler beam with bending stiffness EI, mass m b per unit length, and width B (B 2Δ). The layered ground is formed by N horizontal layers and the underlying half-space. Each of the N soil layers is assumed to be slightly dissipative and is characterized by the shear wave velocity β Li s , the mass density ρ Li s , Poisson's ratio v Li s , and the hysteretic damping ratio ξ Li s (i 1,2,3 . . . N). The underlying half-space is characterized by the complex S-wave velocity β R s , the mass density ρ R s , Poisson's ratio v R s , and the hysteretic damping ratio ξ R s . The rubber-concrete alternating superposition in-filled trench is composed of M rubber-concrete superposition layers. Each of the M rubber-concrete superposition layers is characterized by the shear wave velocity β j t , the mass density ρ j t , Poisson's ratio v j t , and the hysteretic damping ratio ξ j t (j 1,2,3 . . . M). Point P 1 (x 10 m, y 0 m, z 0 m) is the observation point.
When solving the flexibility of the layered elastic foundation with an in-filled trench under the moving load, one section of the vertical track can be used for the solution, and the dynamic response caused by the moving load at the section can be calculated. Then, according to the speed of the train, the remaining cross-sections are shifted to the corresponding phase, and the dynamic response generated at the other cross-sections is obtained. For solving the flexibility of the layered halfspace with in-filled trench under moving load, this paper divides the layered half-space with in-filled trench into two parts: the layered half-space outside the in-filled trench and the soil layer inside the in-filled trench. The total wave field of the layered elastic half-space outside the in-filled trench includes the free field and the scattered wavefield, and the wavefield of the soil layer inside the in-filled trench only includes the scattered wave field. The free wave field is defined as the dynamic response (displacement and stress) of a layered half-space without an in-filled trench under a moving load, which is solved by the direct stiffness method. The scattered wavefield is defined as the additional wavefield generated by the scattering due to the existence of the in-filled trench, which is simulated by applying two sets of virtual moving uniformly distributed line loads on the boundary of the in-filled trench. This method is called the moving Green's function method. The density of the virtual moving uniformly distributed line loads can be determined by the corresponding boundary conditions. Finally, the total wave field of the layered elastic half-space outside the infilled trench is the superposition of the free wave field and the scattered wavefield, that is, the flexibility of the layered elastic half-space containing the in-filled trench under the action of moving uniformly distributed vertical load is obtained.

The Flexibility of the Layered Half-Space with In-Filled Trench
The stiffness matrix given in [29] can be adopted for the dynamic stiffness matrix of the 3D layered foundation [M P,SV,SH ]. If the uniform distributed vertical load moving along the y-axis direction has a distribution length of 2Δ along the x-axis direction, the density is q 0 , which can be expressed as where δ is the Dirac function; ϕ(x) is the distribution function, when |x| ≤ Δ, ϕ(x) 1, otherwise ϕ(x) 0; c is the speed of the moving load. The Fourier transform of Eq. 1 can be obtained as Here, k x and k y are the wavenumbers in the x and y directions, and ω is the circular frequency. In the layered elastic foundation, the external force and displacement on the interface of each soil layer satisfy the equilibrium equation [30] as of the soil layer. Considering that there is only the vertical load on the surface, only the third element has a value, and the other elements are zero. [M P,SV,SH ] is the 3D stiffness matrix of the layered half-space. Substituting Eq. 2 into Eq. 3, the displacement on the interface of each soil layer can be obtained. Subsequently, the amplitude of the upward and downward waves in each soil For the scattered wavefield caused by the in-filled trench, let be the displacement and stress Green's function matrixes in the outer half-space of the infilled trench and [G S u (x, y, z)], [G S t (x, y, z)] be the displacement and stress Green's function matrixes in the inner half-space of the in-filled trench, respectively [31]; then, the displacement and stress of any point outside and inside the in-filled trench can be expressed as Here, the subscripts "u" and "t", respectively, represent the displacement and stress Green's function; the subscript "G" represents the dynamic response caused by the moving uniform distributed line load; the superscripts "D" and "S", respectively, represent the layered elastic half-space outside the in-filled trench and the elastic soil layer inside the in-filled trench.
p 1x ,p 1y ,p 1z T and p 2x ,p 2y ,p 2z T are virtual distributed line load.
The continuous conditions of stress and displacement on the interface s of the in-filled trench can be expressed as Here,T are the stress and displacement along the x-, y-, and z-coordinate directions generated by the free field on each element at the boundary of the in-filled trench. [W(s)] is the weight function matrix, which can be taken as the unit matrix, so that the integral can be performed independently on each element. Substituting Eq. 4 into Eq. 5 can be obtained where From Eq. 6, the virtual moving uniform distributed line loads p 1x ,p 1y ,p 1z T andp 2x ,p 2y ,p 2z T applied on the boundary of the filling trench can be obtained, and substituting them into Eq. 4 can obtain the scattered wave field. Finally, by combining the free wave field and the scattered wavefield, the total displacement of any point in the layered elastic half-space can be obtained ũ x ,ũ y ,ũ z T G ux x, y, z, ω , G uy x, y, z, ω , G uz x, y, z, ω Tq y, ω .
Among them, G u x (x, y, z, ω), G u y (x, y, z, ω), and G u z (x, y, z, ω), respectively, represent the dynamic flexibility coefficient of the layered elastic foundation with in-filled trench under the unit moving uniform distributed line load with a width in the x, y, and z directions;q(y, ω) is the vertical uniformly distributed line load; and w x ,w y ,w z T is the displacement vector generated by the elastic foundation under moving load. The displacement, velocity, and acceleration of any point in the time domain can be obtained by the inverse Fourier transform as shown in the following:

Coupling of the Layered Foundation and the Track
Assuming that the interaction pressure density between the track and the foundation is q(y, t), the vibration equation of the track's vertical displacement Ω can be described as [32]: Here, M n 1 p n (y-ct + L n ) is the wheel load of the train, and its specific parameters can be found in [32]. Equation 15 is converted into the frequency-wavenumber domain to obtain EIk 4 y − Mω 2 Ω k y , ω 2Δ q k y , ω + M n 1 p n k y , ω . (16) Considering that the displacement of each point of the track in the transverse direction is Ω, the continuous condition of the track center displacement and the corresponding foundation point is q k y , ω G w (0, 0, 0, ω) Ω.
Here, G w (0, 0, 0, ω) is the dynamic flexibility of the layered foundation with an in-filled trench under the action of the moving load in the z-direction. It expresses the vertical displacement of the midpoint of the element due to the uniform distributed vertical load acting on the element, which can be obtained by Eq. 13. Combining Eqs 16 and 17 can obtain the interaction force between the track and the layered foundation and the track displacement in the frequencywavenumber domain. Then, the inverse Fourier transform is used to transform the result into the frequency-space domain, and then the displacement response of any position of the elastic foundation in the frequency-space domain can be obtained by Eq. 13. Finally, the displacement, velocity, and acceleration responses of any position of the elastic foundation in the time-space domain are obtained from Eq. 14.

METHOD VERIFICATION
This paper verifies the correctness of the numerical analysis of this paper by comparing the calculated results with the actual measured data of the Swedish X-2000 train [32]. In this section, the material parameters in the in-filled trench are taken as the same as the parameters of the soil layer, that is, the half-space with the in-filled trench degenerates into the halfspace without an in-filled trench. Table 1 and Table 2, respectively, give the foundation soil parameters and the track system parameters of the Swedish X2000 train at different speeds. The Swedish X2000 train consists of five carriages, and the load distribution of the train wheels is shown in Figure 1B. Figure 1C shows the comparison between the calculated results in this paper and the measured data when the train speeds are 70 km/h and 200 km/h, respectively. The results in this paper are in good agreement with the measured data, which confirms the reliability of the method in this paper.

VIBRATION ISOLATION OF THE RUBBER-CONCRETE ALTERNATING SUPERPOSITION IN-FILLED TRENCH
To better demonstrate the superiority of the rubber-concrete alternating superposition in-filled trench for train-induced vibration isolation, this paper takes the vibration generated by the Sweden X2000 high-speed train on the actual soft soil foundation as an example, and compares the vibration isolation effect with that of the empty trench, rubber in-filled trench, and C30 concrete in-filled trench. In the calculation, the thickness of the rubber-C30 superposition unit is taken as 0.4 m, and the trench with 3.0 m depth and 0.5 m width. The actual soft soil foundation parameters of the Sweden X-2000 high-speed train can be found in [33]. The track parameters of the X-2000 train are shown in Table 2, and the wheel axle load distribution is shown in Figure 1B.  Figure 2A shows the results of the vertical displacement, velocity, and acceleration time histories of point P 1 in Figure 1A when the speeds of the X2000 train are 70 km/ h and 200 km/h, with different types of in-filled trench. Figure 2B shows the Fourier amplitude spectrum of the vertical displacement, velocity, and acceleration at point P 1 corresponding to Figure 2A.
It can be seen from Figure 2A that the rubber-filled trench has almost no vibration isolation effect. At the two speeds of the train, the vibration isolation effect of the rubber-concrete alternately superposition in-filled trench proposed in this paper is significantly better than that of the C30 concrete in-filled trench, which is especially reflected in the vibration isolation effect on displacement. Moreover, the vibration isolation effect of the rubber-concrete alternately superposition in-filled trench at the speed of 200 km/h is better than that of the speed of 70 km/h. Figure 2B shows that the frequency spectrum of ground vibration response behind the trench is significantly changed by the rubberconcrete alternately superposition in-filled trench. Compared with low-frequency vibrations generated by the lower train speed, the rubber-concrete alternately superposition in-filled trench has a better vibration isolation effect on high-frequency vibrations generated by a high-speed train. This shows that the rubber-concrete alternately superposition in-filled trench has the frequency band characteristics of elastic waves, that is, elastic waves in the passband frequency range can propagate without attenuation, while the elastic waves in the forbidden frequency range will be filtered out.

CONCLUSION
This paper presented a new train-induced vibration isolation measure of rubber-concrete alternating superposition in-filled trench and built a 2.5D train-track-layered foundation-filled trench model to analyze the dynamics of track and layered foundation with the in-filled trench. The correctness of the model is verified by using the measured data of the Sweden X2000 high-speed train. Finally, the vibration isolation efficiency of rubber-concrete alternating superposition infilled trench is calculated using the actual soft soil foundation parameters of the Sweden X2000 high-speed train, and the vibration isolation efficiency is compared with that of the empty trench rubber in-filled trench and concrete infilled trench. The results show that the vibration isolation effect of the rubber-concrete alternating superposition in-filled trench proposed in this paper is better than that of the C30 concrete in-filled trench, especially the impact on displacement. Compared with low-frequency vibrations generated by the lower train speed, the rubber-concrete alternating superposition in-filled trench has a better vibration isolation effect on high-frequency vibrations caused by higher-speed trains.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.