2.1 Construction of the full-scale experimental model

Three steel spring floating slabs are longitudinally arranged in a 1:1 scale ratio for the experiment. Four sets of side-mounted vibration isolators are installed beneath each floating slab. To create a longitudinally connected floating slab, the three individual slabs are joined by pouring magnesium phosphate concrete (MPC) into the joints between adjacent slabs. The optimal mixture proportion of MPC is from Ref. [28].

WJ-8 fasteners are employed, adhering to the recommended fastening torque range of 140 to 180 N·m according to the Railway Industry Standard of China TB/T 3395.5—2015[29]. For experiment, a torque of 160 N·m is utilized. The fundamental dimensions of the floating slab and the positioning of the steel spring vibration isolators are shown in Figures 1 and 2.

Figure 1全屏查看

MPC location of adjacent floating slabs (Unit: mm)

Figure 2全屏查看

Dimensions and schematic diagram of the full-scale experimental model

2.2 Experimental methods

2.2.3 Test program design

During the drop hammer test conducted on the standalone floating slab, three accelerometers are positioned as follows: one on the ground, one on the surface of the floating slab, and one on the steel rail, as shoun in Figure 3(a). In contrast, the drop hammer test on the longitudinally connected floating slab system involves a setup comprising four accelerometers on the ground and rail tracks, and one accelerometer sensor on the steel rail, as shown in Figure 3(b). The positions and specifications of these sensors remain consistent for both types of rail structures.

Figure 3全屏查看

Test sensor layout: (a) Longitudinally disconnected floating slab; (b) Floating slab with longitudinal connection

3.1 Data evaluation indicators

Before commencing the fatigue test, the analysis focuses on the drop hammer impact test data obtained from the floating slab with a longitudinal connection track system. Figure 4 shows the time domain acceleration vibration curves for #8, #12 and #13, as marked in Figure 3. Figure 4(c) shows that the drop hammer head collided with the rail five times, with an approximate time gap of 0.12 s between the first and second collisions. Following an initial collision lasting approximately 0.07 s, the acceleration of the rail decayed. Conversely, for the floating slab and the ground subjected to impact loading, due to a more gradual decay, the second collision occurred before the acceleration decay from the first collision had fully subsided. Consequently, the acceleration vibration curve for the floating slab and the ground after 0.12 s represents the superposition of vibrations from multiple collisions. Hence, it is reasonable to consider 0.1 s as the data point for analysis.

Figure 4全屏查看

Vibration acceleration time domain curves: (a) Ground; (b) Floating slab; (c) Rail

This paper analyzes the vibration acceleration in the time domain and frequency domain. The time domain analysis indices are the acceleration peak and the root mean square (RMS) of acceleration. The acceleration peak intuitively reflects the peak value of the impact force, and RMS represents the average power of acceleration over a given period. The RMS serves as a method for quantifying the vibration energy of an object, and is computed by extracting the square root of the sum of the squares of the acceleration values at each signal point [35, 36]. A higher vibration energy level corresponds to an elevated RMS value of the vibration acceleration of the structural component. The expression for the weighted RMS of acceleration in the case of a standard continuous function is represented by:

(1)

where a(t) represents the acceleration time-domain curve and T represents the time duration of the acceleration samples.

The main evaluation indices in the frequency domain are the power spectral density (PSD, D_ps), frequency division vibration levels at 1/3 octave, the vibration acceleration level (VAL, L_va), and the vibration attenuation (VA, V_a). When subjected to a drop hammer impact, the PSD distribution of the ground, floating slab, and rail at the excitation point cross-section is obtained by performing a Fourier transform of the time domain curve to obtain its frequency domain signal. Through the analysis of the PSD, the distribution law of the energy spectral density of vibration in different positions of each part of the track structure is obtained. According to Ref. [11], the average power spectral density of a random digital signal is typically characterized as having limited average power as follows:

(2)

where and T denote the output and input signal durations of the fast Fourier transform (FFT) of the random vibration signal, respectively, and T is taken as 0.1 s.

The collected acceleration data are transformed from the time domain to the frequency domain through a fast Fourier transform (FFT) and 1/3 octave analysis. In the problem of subway vibration, the vibration analysis of low-frequency bands is very important. The 1/3 octave analysis is based on the octave, and the frequency band is further refined. Following the requirements of the technical specification for floating slab track CJJT—191—2012 [33], the RMS difference in the frequency range of 1-1000 Hz between the floating slab track and the normal track slab for vertical ground vibration is used as an indicator to evaluate the vibration reduction effect of the floating slab track, with the maximum difference in the frequency range being the reference quantity. Additionally, to compare the overall vibration level of different track structures in the vertical direction, the vibration acceleration grade is introduced. The vibration acceleration level (L_va) is calculated according to Formula (3) by selecting the data in the frequency range of 1-1000 Hz with significant influence as follows:

(3)

The vibration decay rate per linear meter (V_a) at varying distances from the excitation point and is expressed as follows:

(4)

where is the difference in the vibration acceleration grade between two points and is the distance between the two points.

3.2 Time-domain analysis of vibration characteristics

Figure 5 shows the time-domain plots illustrating the vibration acceleration on the ground, the floating slab and the rail under impact loading conditions for both Case 1 and Case 2. Compared with Case 1, Case 2 exhibits a minor increase in the acceleration amplitude on the rail and floating slab but a significant reduction in the ground acceleration amplitude. The longitudinally connected floating slab system is beneficial to the reduction of the impact load on the foundation and enhancement of vibration attenuation.

Figure 5全屏查看

Comparison of the time-history curves between Case 1 and Case 2: (a) Ground; (b) Floating slab; (c) Rail

Figure 6 presents the time-domain plots illustrating the vibration acceleration on the ground, the floating slab, and the steel rail under impact loading for Case 2, Case 3, and Case 4. The data in the figure show that fatigue loading exacerbates the acceleration amplitude on the ground while diminishing the acceleration amplitude on the steel rail and the floating slab.

Figure 6全屏查看

Comparison of the time-history curves between Case 2, Case 3 and Case 4: (a) Ground; (b) Floating slab; (c) Rail

The graphed data illustrate variations in both acceleration peak values and RMS values relative to Case 1. As shown in Figure 6(a), on the ground, the peak acceleration and RMS change by -37.3%, -31.4%, -25.5% and -35.6%, -29.8%, -26.0% for Case 2, Case 3 and Case 4, respectively, relative to Case1. Case4 exhibits an increase of 18.8% and 14.9% compared to Case 2.

As shown in Figure 6(b), on the floating slab, the acceleration peak values and RMS values for Case 2, Case 3 and Case 4 compared to Case 1 are 68.9%, 36.9%, 22.1% and -15.6%, -18.5%, -20.1%, respectively. However, Case 4 records decrease by -27.8% and -5.3% relative to Case 2. This contrast arises from Case 1’s lower peak accelerations, coupled with its slower overall vibration attenuation, resulting in an increased RMS value.

As shown in Figure 6(c), on the rail, the acceleration peak values and RMS values for Case 2, Case 3 and Case 4 relative to Case 1 show significant shifts: 82.9%, 53.1%, 14.3% and 31.8%, 10.1%, -16.2%, respectively. Case 4 experiences notable declines of -37.5% and -36.5% relative to Case 2.

Combining Figures 7 and 8, it can be seen that the new structure using the MPC connection has a positive effect on reducing the vibration on the ground, whereas the peak vibration acceleration and RMS on the rail and the floating slab correspondingly increase, which is in accordance with the law of conservation of energy. Fatigue loading also affects the damping effect of the structure, as the fatigue loading increases, the peak acceleration and RMS at the ground also increase, but the structure after fatigue testing still has better damping effect compared to a single floating slab.

Figure 7全屏查看

Track subsystems acceleration peak for Case 1, Case 2, Case 3 and Case 4

Figure 8全屏查看

Track subsystems acceleration RMS for Case 1, Case 2, Case 3 and Case 4

3.3 Frequency domain analysis of vibration characteristics

For the same conditions, the data from each analysis are averaged to produce the results of the analysis. The results of the PSD for the rail, floating slab, and ground are shown in Figures 9 and 10.

Figure 9全屏查看

Power spectrum curves for Case 1 and Case 2: (a) Ground; (b) Floating slab; (c) Rail

Figure 10全屏查看

Power spectrum curves for Case 2, Case 3 and Case 4: (a) Ground; (b) Floating slab; (c) Rail

Figure 9 shows the power spectra within the 1000 Hz frequency range. The peaks of the longitudinally disconnected floating slab are significantly larger than those of the longitudinally connected floating slab with the track. For the ground, the longitudinally disconnected floating slab exhibits its first peak at approximately 11 Hz, a second peak at approximately 96 Hz. In contrast, the longitudinally connected floating slab displays its first peak at approximately 10 Hz, followed by multiple more pronounced peaks ranging from 65 to 149 Hz. For the floating slab, both floating slab structures exhibit multiple peaks within the 100-800 Hz range, closely spaced near on the rail, with a maximum peak magnitude approximately 6.5 times greater for the longitudinally disconnected floating slab than for the longitudinally connected floating slab. The power spectrum of the longitudinally connected floating slab structure is more concentrated and exhibits lower vibration levels overall, indicating a more reasonable energy distribution. Figure 10 demonstrates that, under varying fatigue loads, the peak positions of the power spectrum remain consistent, with differences observed in the peak values. The ground power spectrum peaks increase with fatigue cycles, while those on the floating slab decrease. On the rail, peak values vary, but the maximum peak appears in the power spectrum line before fatigue cycles, suggesting an increase in the maximum value with increasing fatigue cycles. The impact load experiences minor changes within the track structure but substantial variations in energy magnitude. The energy on the rail slightly decreases, that on the floating slab decreases, and the energy transferred to the ground increases. Multiple peaks below 100 Hz were observed on both the ground and the floating slab, while no such peaks were present on the steel rails, which was attributed to the natural frequency of the floating slab falling below 100 Hz, thereby amplifying some of the vibration energy within certain frequency ranges.

To compare the vibration transmission and attenuation of different track structures under impact loading, the frequency division vibration levels at 1/3 octave from 1 to 1000 Hz were calculated for the components of the two structures, as shown in Figures 11 and 12.

Figure 11全屏查看

Frequency division vibration levels of Case 1 and Case 2: (a) Ground; (b) Floating slab; (c) Rail

Figure 12全屏查看

Frequency division vibration levels of Case 2, Case 3 and Case 4: (a) Ground; (b) Floating slab; (c) Rail

Over a frequency range of 1-1000 Hz, the vibration propagates along the steel rail-floating slab-ground path, as depicted in Figure 13. Within this spectrum, there is a progressive reduction in vibration acceleration levels at various center frequency points for both rail structures, due to the effective vibration isolation properties of the fasteners and isolators.

Figure 13全屏查看

VAL for Case 1, Case 2, Case 3 and Case 4

In Case 1, the frequencies with the minimum and maximum deviations between the orbit and the ground are 12.5 and 250 Hz. In Case 2, these frequencies are 10 and 125 Hz, respectively. Within the 1-5 Hz range, Case 1 exhibits significantly higher vibration levels than Case 2, with the maximum deviation from the ground occurring at 2 Hz, measuring 18.1 dB. In the 5-50 Hz range, both cases display nearly identical acceleration vibration levels, with Case 2 slightly lower than Case 1. However, beyond 50 Hz, there is a distinct alternating distribution between the two, indicating that the longitudinally connected floating slab structure effectively mitigates low-frequency ground vibrations. For the vibration distribution on the floating slab, there is a distribution pattern similar to that of the ground. This study underscores the effectiveness of longitudinally connected floating slab structures in reducing low-frequency vibrations on both the ground and floating slab track.

Figure 12 shows the fundamental spectrum of the 1/3 octave frequency bands at different fatigue cycles. The primary distinctions are mainly observed in the lower-frequency region, and there are minimal differences in the higher-frequency region. The spectral vibration levels below 6.3 Hz increase with increasing number of fatigue cycles. As shown in Figure 12(a), on the ground, both Case 3 and Case 4 exhibit their maximum differences in spectral vibration levels at 3.15 Hz, measuring 5.1 and 6.7 dB, respectively.

Significant differences in spectral vibration levels below 5 Hz are noted on the floating slab, and these differences increase with increasing fatigue cycles. The maximum disparities in spectral vibration levels between Case 3 and Case 4 occur at 1.25 and 1.6 Hz, measuring 5.3 and 16.5 dB, respectively. The acceleration spectral vibration levels at the steel rail locations remain relatively constant with an increase in the number of fatigue cycles. The differences in the vibration levels at various center frequencies are all below less than 10 dB. Consequently, it is reasonable to conclude that under fatigue loading, the acceleration vibration levels at the steel rail locations remain largely unchanged.

Figure 13 illustrates the vibration acceleration levels of the steel rails, floating slab, and ground under different scenarios.

Figure 13 illustrates the reductions in the levels of vibrations on the ground, floating slab, and steel rail in Case 2 relative to Case 1 within the cross-sectional area of the board; reductions of 1.1%, 7.0%, and 5.6% were observed, respectively. This improvement is due to the longitudinally connected floating slab track structure, which effectively distributes vibrational energy across multiple floating slab structures and longer sections of the steel rail, significantly reducing the energy levels at impact points. With the increasing number of fatigue cycles, the vibration acceleration levels on the ground increases, in increments of 1.6% and 4.4% after 5×10⁶ and 10⁷ cycles, respectively, compared to those under prefatigued conditions. Conversely, the vibration acceleration levels on the floating slab and steel rail decrease with an increasing number of fatigue cycles. After 5×10⁶ and 10⁷ cycles, they decrease by -0.03% and 1.2%, and 0.04% and 0.03%, respectively, relative to the prefatigued state. Fatigue loading leads to an increase in the stiffness of the isolators, reducing the damping effectiveness of the floating board.

Figure 14 shows the the VAL at different sensor locations. The horizontal axis scale is consistent with the distance between sensors. Figure 15 shows the VA at different distances. It illustrates that the vibration acceleration levels at each measurement point on the ground increase with the number of fatigue loadings, while on the floating slab, the vibration acceleration levels decrease with the number of fatigue loadings. The vibration acceleration decays nonlinearly along the longitudinal direction on both the ground and the floating slab.

Figure 14全屏查看

Vibration acceleration levels at different measurement points: (a) Ground; (b) Floating slab

Figure 15全屏查看

VA at different distances: (a) Ground; (b) Floating slab

As shown in Figure 15(a), before the fatigue test, the average longitudinal VA is 0.88 dB/m in the range of 0-1200 mm from the excitation point on the ground, and decreases to 0.60 dB/m at the joint location of the MPC connection; after the fatigue test, the average longitudinal VA is 0.83 dB/m in the range of 0-1200 mm from the excitation point. Following the MPC (1200-3600 mm from the excitation point), the average longitudinal VA decreased to 0.21 dB/m, while it was 0.31 dB/m in the range of 3600-6000 mm from the excitation point. As shown in Figure 17(b), the average longitudinal VA decreased to 0.12 dB/m at the MPC position from the surface of the floating slab before the fatigue test; after the fatigue test, the average longitudinal VA at the same position decreased to 0.31 dB/m.

After applying 10⁷ fatigue loads, the VA of the ground at the MPC position decreased by 65.6% with increasing fatigue load, the fatigue effect causes the stiffness of the steel spring isolator on both sides of the joint to increase significantly compared with the stiffness of the isolator at other locations. As a result, there is a reduction in the vibration acceleration energy difference between the ground below the joint and the ground below the excitation point, leading to a decrease in the vibration attenuation of the ground below the joint. Conversely, the VA of the floating slab near the excitation point increases by 171.9% with increasing fatigue load. This is caused by the fact that the fatigue loading leads to a reduction in strength at the joint between the MPC and the floating slabs. Although this leads to a reduction in the overall structural capacity, it also enhances the structure’s ability to dissipate longitudinal vibration energy. Consequently, there is an increase in the vibration attenuation at the MPC location on the surface of the floating slab.

The results show that the ability of MPC to dissipate vibration energy increases as the number of fatigue loads increases. As shown in Ref. [28], compared to ordinary silicate concrete (OPC), MPC has lower porosity when connected to OPC, which results in higher joint strength and improves longitudinal transmission of vibration; as the vibration transmission capacity of the MPC decreases, the strength of the interface connecting MPC to OPC subsequently decreases. To verify this conclusion, static load tests are conducted before and after the fatigue experiments. A static load is applied at the MPC position, and concrete resistance strain gauges are used to measure the strain at the MPC and OPC connection positions. The strain gauges are located at the bottom of the connection posit, which is the most vulnerable area. Figure 16 shows that, there is a significant increase in the strain at the same location under the same load, which is consistent with the results of our previous analysis.

Figure 16全屏查看

Load force-strain curves at the MPC and OPC connection positions