Ground vibration mitigation by using ballast mat

2.1. Train load

Wheel-rail forces mainly appear in 3 frequency ranges: (1) relative motion of the train body and suspension in 0.5 Hz-10 Hz; (2) It is caused by the spring-back effect of the mass of wheelset on the rail in intermediate frequency range (30 Hz-60 Hz); (3) It is produced by the resistance of the wheel-rail contact surface due to the movement of the rail in high frequency range (100 Hz-140 Hz) [6].

Therefore, it is feasible to simulate the train load with an excitation force function, as shown in Eq. (1):

where $k_1$, $k_2$ are the superposition and dispersion coefficient, $P_0$ is the static train load; $P_1$, $P_2$, $P_3$ are the loads related to the above three frequency ranges.

Let the unsprung mass of the train be $M_0$, the load can be defined by Eq. (2):

The relevant parameters of the train are shown in Table 1. According to the calculation conditions, the selected A-type subway train load simulation equation is Eq. (3):

2.2. Track model

The track model consists of steel rails, sleepers and ballast. The track is U75V hot-rolled steel rail, and 60 kg/m. The rail gauge is 1435 mm; the track bed is a gravel track bed with a thickness of 0.45 m. Table 2 shows the parameters for track model.

2.3. Soil model

The soil model dimension is selected as 130 m × 130 m × 55 m, which is shown in Fig. 2. The vibration caused by the subway train is in a micro vibration, thus the soil model is adopt the linear elastic model. The soil property can be obtained from the results of the geological survey.

Fig. 1Finite element model of track and sleeper

Fig. 2Foundation soil model

2.4. Ballast mat

A ballast mat is set under the ballast. The ballast mat is made of high-quality rubber, wear-resistant, oil-resistant, and ozone-resistant (Fig. 3). Its width is 1500 mm. Its thickness is 27 mm, and the unit area weight is 18 kg/m². When the load is 0.020-0.100 N/mm², the static foundation modulus is 0.030 N/mm³±15 %, and when the preload is 0.060 N/mm² and 40 Hz, the dynamic foundation modulus is 0.080 N/mm³±15 %. It is assumed that the fixed hysteretic damping value is 0.1 for the mat. Fig. 4 shows the finite element model after adding ballast mat. Ballast mat was inserted between the sub-ballast and embankment.

Fig. 3Ballast mat

Fig. 4Finite element model

2.5. Model verification

Set the train model to Type A as the load input parameter, compare the calculation results for different distance from the track of the numerical model with field measurement data, and the reliability of the numerical model is verified, as shown in Fig. 5. It can be seen that the calculated value matched well with the measured value, which indicated the established numerical model can better simulate the vibration damping characteristics of ballast mat.

Fig. 5Model verification

Fig. 6Ballast mat vibration reduction effect

3.1. Influence of setting ballast mat

The vibration mitigation characteristics of setting ballast mat are shown in Fig. 6. It can be seen that the ballast mat can effectively attenuate the vibration. The ground vibration can be reduced by about 10 dB within 30 m from the center line of the track, and the attenuation of vibration decreases with the increase of the distance from the center line of track.

3.2. Impact of the thickness of the ballast mat on vibration reduction effect

In order to know the influence of the thickness of the ballast mat on vibration mitigation, the thickness of the ballast mat was changed without changing the width and the static foundation modulus, and combined with the actual situation of the ballast mat model, 15 mm, 23 mm, 27 mm and 32 mm are compared and analyzed.

In Fig. 7, the vibration isolation effect of the ballast mat is obvious as the thickness of the ballast mat increases. However, the improvement of the vibration mitigation is not obvious when the thickness is enough. The closer to the track centerline, the increase in the thickness of the ballast mat has a more obvious impact.

Fig. 7Vibration mitigation of ballast mats of different thicknesses

Fig. 8Vibration mitigation of ballast mats under different static foundation modulus

3.3. Impact of the static foundation modulus of ballast mat

To study the effects of the static foundation modulus of the ballast mat on the vibration mitigation, the static foundation modulus of the ballast mat was changed without changing the width and thickness, combined with the actual production situation of the ballast mat, respectively, as shown in Fig. 8.

As the static foundation modulus increases, the vibration mitigation of the ballast mat increases, especially as the closer to the center line of the rail, the increase effect is more obvious. However, at some distance from the center line of the rail, the vibration isolation effect tends to be consistent.