Influence of unsupported sleepers on the dynamic stability of ballasted bed based on wheelset impact tests

The unsupported sleeper can change the load characteristics of ballast particles and thus affect the dynamic stability of a ballasted bed. In this work, a laboratory test was constructed on a ballasted track containing unsupported sleepers. The ballasted track was excited by a wheelset, and the influence of unsupported sleepers on the dynamic stability of a ballasted bed was studied. The results show that the main frequency of the sleeper vibration appeared at 670 Hz, and the first-order rigid vibration mode at the frequency of 101 Hz had a significant effect on the condition without the unsupported sleeper. When the sleepers were continuously unsupported, the vibration damping effect of ballasted bed within the frequency range of 0–450 Hz was better than that at higher frequencies. Within the frequency range of 70–250 Hz, the vibration damping effect of the ballasted bed with unsupported sleepers was better than that without the unsupported sleeper. Owing to the excitation from the wheelset impact, the lateral resistance of the ballasted bed with unsupported sleepers whose hanging heights were 30, 60, and 90 mm increased by 37.43%, 12.25%, and 18.23%, respectively, while the lateral resistance of the ballasted bed without the unsupported sleeper remained basically unchanged. The unsupported sleeper could increase the difference in the quality of the ballasted bed between two adjacent sleepers. In addition, test results show that the hanging height of the unsupported sleeper had little effect on the lateral resistance of a ballasted bed without external excitation, but had an obvious effect on the rate of change of the lateral resistance of a ballasted bed and the acceleration amplitude of the sleeper vibration under the wheelset impact.


Introduction
A ballasted track features good elasticity, a low construction cost, and ease of maintenance [1,2]. As one of the main components of the ballasted track structure, the sleeper serves mainly to withstand the force from the steel rail, to transmit the load to the ballasted bed evenly, and to maintain the geometry of the track. Owing to the unevenness of the rail surface and the uneven support rigidity of the ballasted bed [3], the ballasted bed is prone to uneven settlement under the long-term effect of the train load. The sleeper can even lose contact with the ballasted bed and become unsupported. As the unsupported sleeper can change the force characteristics of the ballast particles and the stiffness of the support under the sleeper, the wheel-rail interaction force and the ballasted bed settlement increase when the train passes over the section of the unsupported sleeper, which can change the dynamic response of the ballasted track, aggravate the deterioration of the sleeper, fastener system, and ballast particles, and seriously affect the operation safety of trains [4][5][6]. For this reason, railway maintenance departments always regard the prevention of unsupported sleepers as a key task of track management. In the actual operation of a railway, sleepers are mostly partially unsupported. Based on the actual track maintenance experience of the railway maintenance departments, we know that the unsupported locations are mainly at the ends and in the middle of sleepers, as shown in Fig. 1.
The sleeper unsupported on one end poses a greater risk to the operation safety of trains as compared to the sleeper unsupported in the middle. For this reason, we mainly studied sleepers unsupported on one end in this paper. The unsupported sleeper significantly increases the vibration of the sleeper boxes of adjacent sleepers on both sides and the ballast particles underneath, which can accelerate the degradation of the ballast particles [7]. In addition, with an unsupported sleeper, the compaction state of the ballasted bed is different from that in the normal state, and the difference in the compaction state has a greater impact on the stability of the ballasted bed [8]. Therefore, unsupported sleepers have a serious impact on the dynamic stability of a ballasted bed, which is worth studying.
The dynamic test can directly obtain the vibration response of the object to be tested, and it is usually completed with instruments such as a dynamic test system and acceleration sensors. Liu et al. [9] used an IMC dynamic test instrument and ballast sensors to test the vibration responses of sleepers and ballast particles at the sleeper ends, ballast shoulders, and sleeper boxes. They studied the transmission characteristics of the ballast vibration on the surface of a ballasted bed during shock excitation and successfully monitored the health status of the ballasted track. The results show that when the ballasted bed was excited at a high frequency, the ballast on the surface at the sleeper boxes tended to splash and pose a serious threat to the health of the track. Hu et al. [10] tested the dynamic response characteristics of a ballastless track structure in the bridge-subgrade transition section using an IMC dynamic test instrument and acceleration sensors. They found that the dynamic response of the track structures increased significantly when the train's speed was greater than 275 km/h. Moreover, Ma et al. [11] compared the vibration damping effect of trapezoidal sleepers with that of ordinary sleepers in ballasted track structures through dynamic tests. They found that the vibration damping effect of the ballasted bed with trapezoidal sleepers was better, which could effectively control environmental vibrations at a low frequency. Because the wheelset impact test can cause significant excitations, it is often applied to test the dynamic characteristics of track structures over a wide frequency range [12]. For this reason, we chose to adopt the wheelset impact test to conduct the vibration test of the ballasted bed with and without the unsupported sleeper in this study.
Researchers globally have carried out extensive studies on the stability of a ballasted bed. For example, Guo et al. [13] studied the influence of nail length and nail number on the lateral resistance of a ballasted bed by actual measurements and simulations analysis. They found that compared with ordinary sleepers, nailed sleepers could improve the lateral resistance by 53.7% when the nail length was 400 mm. Liu et al. [14] and Zeng et al. [15] investigated the correlation between the lateral and longitudinal resistance of a ballasted bed by carrying out actual measurements. The test results show that there was a strong positive correlation between the lateral resistance and longitudinal resistance of a ballasted bed, and they were positively correlated with the quality state of the ballasted bed. Khatibi et al. [16] established a discrete element analysis model of a ballasted bed and studied the factors affecting the lateral resistance of the ballasted bed. Their results show that the porosity of a ballasted bed was the most direct factor that affected the lateral resistance of the ballasted bed. When the porosity of a ballasted bed decreased by 10%, the lateral resistance of the ballasted bed increased by 18% on average. They obtained a positive correlation between the lateral resistance and quality of the ballasted bed by actual measurements and simulations. Moreover, Jing et al. [17] studied the effect of the unsupported sleeper on a ballasted bed using the discrete element  method and concluded that the lateral resistance of a ballasted bed could be used as an indicator to evaluate the unsupported status of sleepers, and the unsupported sleeper could significantly reduce the lateral resistance of a ballasted bed. Sadeghi et al. [18] built a three-dimensional numerical model of a ballasted track to investigate the effects of the unsupported sleeper on the dynamic behavior of a ballasted track from the seat load, sleeper bending moment, and sleeper-ballast contact force. Sysyn et al. [19] studied the mechanism of the sleeper-ballast dynamic impact and residual settlements caused by impact and quasi-static loading when the sleeper was unsupported. All the above studies showed that the lateral resistance could be used to evaluate the quality of a ballasted bed and indirectly reflect the unsupported state of sleepers.
In this work, a test model of a ballasted track in the laboratory was built, and different hanging height of sleepers were set up. The ballasted track was excited by a wheelset impact, and we acquired the signals of the sleeper vibration by the IMC dynamic test instrument and investigated the signals in time domain, frequency domain, and transmission characteristics. Moreover, we used hydraulic jacks to test the lateral resistance of the ballasted bed before and after the wheelset impact and analyzed the lateral resistance of the ballasted bed from the perspectives of the variety and transmission characteristics.

Test model
A full-scale indoor test model of a ballasted track was built in the laboratory using the grade-1 ballast, type III concrete sleepers, CHN60 steel rails, and type II elastic bar fastener systems. The thickness of the ballasted bed was 0.35 m, the side slope of the ballasted bed was 1:1.75, and the pile height of the ballasted shoulder was 0.15 m. After the ballasted bed had been built, we tamped, compacted, and stabilized it. The volume density of the ballasted bed exceeded 1.70 g/cm 3 , which was measured by the irrigation method.
When setting the unsupported sleeper, we first dug downwards from the position of the sleeper end to the hanging height specified in Table 1 for the various test conditions. Then, the void was filled with an ice box of equal volume to prevent the ballast particles from entering the unsupported area during the tamping and stabilizing process. The ice box was approximately a taper, whose diameter of the bottom surface was nearly 24 cm and height was the hanging height of the unsupported sleeper (30, 60, and 90 mm). Before the ice box melted, the ballast was filled at the sleeper end position to restore the ballasted bed to its initial dimensions and ensure that the volume density of the ballasted bed was above 1.70 g/cm 3 through tamping, compaction, and stabilizing. The tests were carried out after the ice box had melted and the water had evaporated completely.
All the sleepers in the test model were set up with the hanging heights specified for their corresponding test conditions, as listed in Table 1. In order to reduce the boundary effects of the model on the test results, we selected the middle four sleepers as the test objects, as shown in Fig. 2.

Test conditions
We used the hanging height as the variable for the test conditions to carry out studies on sleepers unsupported on one end. Referring to the related studies [20,21], the maximum hanging height of the sleeper is about 60 mm, so we conducted comparison experiments with the hanging heights set to 0, 30, 60, and 90 mm. The dynamic stability differences of the ballasted bed for the unsupported sleeper with different hanging heights were studied through the test conditions listed in Table 1.

Test method
When performing the wheelset impact test, the support of the reaction frame was fixed on the concrete surface on both sides of the ballasted bed. Using high-strength electromagnets, we positioned the wheelset above the steel rail. Then, the accelerometer base was adhered to the center point of the upper surface of the sleeper to be tested with a high-strength AB adhesive. After the base had been securely attached, the accelerometer was installed on the base, and then the IMC was connected to the accelerometer with signal lines. During the test, the wheelset fell vertically to impact the ballasted track, and the acceleration data of the sleeper were collected by the eight-channel IMC when the electromagnet was turned off. In order to fully excite the vibration of each part of the track, we dropped the wheelset from a height of 20 mm [22], which means that the wheelset was at a vertical height of 20 mm from the steel rail. The mass of the wheelset was 1120 kg, the range of the accelerometer was 20 g, and the sampling frequency was 20,000 Hz. In the data acquisition process, if the time difference between the left wheel touching the left rail and the right wheel touching the right rail was less than 3 ms, it was recorded as a valid wheelset impact [23], as shown in Fig. 3.

Analysis of the sleeper vibration in time domain
To reduce the boundary effect on the vibration response of the sleeper, we chose sleeper #2 of Fig. 2 as the test object and acquired the time curve of the acceleration for each test condition listed in Table 1, as shown in Fig. 4. The results in Fig. 4 show that there were two pronounced impact effects by the wheelset on the track. This is because the wheelset and the steel rail had a certain degree of elasticity, which caused the secondary impact. With the excitation of the wheelset impact, the acceleration amplitude of sleeper vibration was the largest in test condition 4, at 29.01g, and was the smallest in test condition 1, at 24.10g. The greater the hanging height of the sleeper, the fewer ballast particles at the bottom of the sleeper, and the weaker the vibration damping effect when the sleeper vibrated. As a result, with the hanging height of the unsupported sleeper increasing, the vibration response amplitude of the sleeper gradually increased. As the train

Analysis of the sleeper vibration in frequency domain
To eliminate small fluctuations in the signals acquired by the accelerometer, we processed the signal of the sleeper vibration in time domain with an infinite impulse response (IIR) high-pass filter and a Butterworth filter [24]. Through the fast Fourier transform (FFT) [25], the signal of the sleeper vibration in time domain was converted to the signal in frequency domain, as shown in Fig. 5.
Within the range of 0-1000 Hz, the acceleration curves of the sleeper vibration in frequency domain for all test conditions exhibited two pronounced peaks. The data show that the frequency of the first peak was at 98 Hz for the condition without the unsupported sleeper, while the frequencies of the first peaks for sleepers with 30, 60, and 90 mm hanging heights were 90, 79, and 73 Hz, respectively. The second peak appeared at a frequency of 670 Hz, which is the main frequency of the sleeper vibration.
According to the study by Aikawa [26], based on the coupled vehicle-track dynamic system, the frequency of the first-order rigid vibration mode of the sleeper is as follows: where f 1 is the frequency of the first-order rigid vibration mode of the sleeper, m is the mass of the sleeper, and k is the support stiffness under the sleeper. According to Eq. (1), the first-order rigid vibration mode of the sleeper has the same frequency as the natural frequency of the undamped system with a single degree of freedom. To study the effect of the natural frequency of the sleeper on the forced vibration, we assumed that the mass of the sleeper was 300 kg, and the support stiffness under the sleeper was 120 kN/mm, in accordance with the Chinese code for design of high-speed railway (TB10621-2014). Therefore, the frequency of the first-order rigid vibration mode of the sleeper was about 101 Hz, which is close to the frequency of the first peak at 98 Hz in test condition 1, as shown in Fig. 5. This indicates that the frequency of the first-order rigid vibration mode of the sleeper has a significant effect on the condition without the unsupported sleeper. Because the unsupported sleeper changes parameters such as the thickness of the ballasted bed and the support stiffness under the sleeper, its firstorder rigid vibration mode frequency changes. As a result, there was no obvious peak at 98 Hz in test conditions 2-4, as shown in Fig. 5.

Analysis of transmission characteristics of sleeper vibration
To study the effect of unsupported sleepers on the vibration transmission characteristics of a ballasted bed, we chose sleeper #1 and sleeper #2 of Fig. 2 as the test objects. The vibration of sleeper #2 was regarded as the input to the system, and the vibration of sleeper #1 was regarded as the output of the system. The acceleration transmission rate of the sleeper vibration was defined as follows [27]: where ATR is the acceleration transmission rate of the sleeper vibration and H(A n , A) is the amplitude of the transfer function for output A n and input A.
Based on Eq.
(2), we obtained the acceleration transmission rate between sleeper #2 and sleeper #1 for each test condition listed in Table 1, as shown in Fig. 6.
The results displayed in Fig. 6 show that within the frequency range of 0-1000 Hz, the acceleration transmission rate of sleeper vibration in test condition 1 was less than zero the entire time, which indicates that the sleeper in test condition 1 has a significant vibration damping effect in the frequency range of 0-1000 Hz. For test conditions 2-4, the acceleration transmission rates of sleeper vibration were less than zero in the frequency range of 0-450 Hz, which indicates that when the sleepers are continuously unsupported, the vibration damping effect of the ballasted bed within the frequency range of 0-450 Hz is better than that at higher frequencies. Within the frequency range of 70-250 Hz, the acceleration transmission rate of sleeper vibration with unsupported sleepers was smaller than that without the unsupported sleeper, reflecting that the vibration damping effect of the ballasted bed with unsupported sleepers is better than that without unsupported sleepers in this frequency range. When the frequency was greater than 450 Hz, the acceleration transmission rates of the sleeper vibration in test conditions 2-4 were more than zero, indicating that the vibration amplitude of the sleeper is amplified along the longitudinal direction of the railway within the frequency range of 450-1000 Hz and that the vibration damping effect of the ballasted bed decreases. The train load is a broad-frequency excitation for the ballasted bed [9], and the unsupported sleeper can play a great vibration damping role in the frequency range of 70-250 Hz. However, it causes a high-frequency excitation in the frequency range of 450-1000 Hz, which can not be absorbed by the ballasted bed, and then aggravates the vibration of the sleeper and accelerates the deterioration of

Analysis of the variety characteristics of the lateral resistance of a ballasted bed
The lateral resistance of a ballasted bed is an important indicator for evaluating the stability of the ballasted bed and a key factor for maintaining the stability of a railway [28]. We chose sleeper #2 of Fig. 2 as the test object and measured the lateral resistance of the ballasted bed with a hydraulic jack. Following the design specification of a seamless railway track [29], the lateral resistance of a ballasted bed was taken as the resistance value when the lateral displacement of the sleeper was equal to 2 mm. We analyzed the changes in the lateral resistance before and after the wheelset impact on the ballasted track for the various test conditions listed in Table 1 and defined the rate of change of the lateral resistance of a ballasted bed as follows: where T is the rate of change of the lateral resistance of a ballasted bed, R a is the lateral resistance measured after the wheelset impact, and R b is the lateral resistance measured before the wheelset impact. This rate of change indicates the increase in the lateral resistance of the ballasted bed due to the excitation from the wheelset impact, which reflects the dynamic stability of the ballasted bed. The lateral resistance of the ballasted bed and its rate of change for the different test conditions under the wheelset impact were acquired, as shown in Fig. 7. Figure 7 shows that the lateral resistance of a ballasted bed was the largest in test condition 1. Compared with test condition 1, the lateral resistance of the ballasted bed in test conditions 2-4 decreased by 32.35%, 30.68%, and 35.12%, respectively, which indicates that there is a significant difference in the lateral resistance of a ballasted bed with and without the unsupported sleeper. The stability of the ballasted bed with the unsupported sleeper is poorer than that without one. For all test conditions, the lateral resistance of a ballasted bed increased under the wheelset impact; that is, the stability of a ballasted bed improved to a certain extent owing to the excitation from the wheelset impact. The increase in the lateral resistance of the ballasted bed was the smallest in test condition 1, at 4.25%, and was the greatest in test condition 2, at 37.43%. The lateral resistance of the ballasted bed was increased by 12.25% and 18.23% in test conditions 3 and 4, respectively. Because the increase in the lateral resistance of the ballasted bed was less than 5% in test condition 1, it could be considered that there was no change in the lateral resistance of the ballasted bed before and after the wheelset impact for test condition 1 [30], indicating that the external impact has little effect on the ballasted bed without the unsupported sleeper. In contrast, for test conditions 2-4, when the unsupported sleepers were present, the volume density and stability of the ballasted bed were low. Owing to the excitation from the wheelset impact, the volume density of the ballasted bed increased, which significantly increased the lateral resistance of the ballasted bed.

Analysis of transmission characteristics of the lateral resistance of a ballasted bed
To study the effect of unsupported sleepers on the dynamic stability of a ballasted bed, we chose sleepers #1 and #2 of Fig. 2 as the test objects. The lateral resistance of sleeper #2 was regarded as the input to the system, and the lateral  Fig. 7 Variety characteristics of the lateral resistance of a ballasted bed: a lateral resistance curve; b rate of change of the lateral resistance resistance of sleeper #1 was regarded as the output of the system. The transmission rate of the lateral resistance of a ballasted bed was defined as follows: where D is the transmission rate of the lateral resistance of a ballasted bed and R 2 and R 1 are the lateral resistances of the ballasted bed measured by sleeper #2 and sleeper #1, respectively. The transmission rate reflects the attenuation characteristics of the lateral resistance of the ballasted bed along the longitudinal direction of a railway. If the transmission rate of the lateral resistance of a ballasted bed increased owing to the excitation from the wheelset impact, the difference in the stability of the ballasted bed between two adjacent sleepers had increased. If the transmission rate decreased, the difference in the stability of the ballasted bed between two adjacent sleepers had decreased. The transmission rates of the lateral resistance of the ballasted bed between sleeper #2 and sleeper #1 before and after the wheelset impact for each test condition listed in Table 1 are shown in Fig. 8.
The results in Fig. 8 show that the transmission rate of the lateral resistance of the ballasted bed decreased owing to the excitation from the wheelset impact in test condition 1, indicating that under external excitation, the difference in the stability of the ballasted bed without the unsupported sleeper between two adjacent sleepers had decreased, the quality of the ballasted bed was good, and the dynamic stability was high. In contrast, the transmission rate of the lateral resistance of the ballasted bed in test conditions 2-4 increased, which indicates that for the ballasted bed with continuous unsupported sleepers, the attenuation of the lateral resistance between adjacent unsupported sleepers increases along the longitudinal direction of the railway under external excitation, which leads to a large difference in the stability of the ballasted bed between two adjacent sleepers. When the train passed over the unsupported sleepers, the performances of the ballasted bed at two adjacent sleepers differed greatly, which could result in serious difference in the stress between two adjacent sleepers, aggravate the track irregularity, and affect the normal service state of the ballasted bed.

Conclusions
On the basis of the wheelset impact tests, we studied the effect of unsupported sleepers on the dynamic stability of a ballasted bed from the characteristics of the vibration response of the sleepers and the lateral resistance of a ballasted bed. The major findings of this paper can be summarized as follows: (1) The main frequency of the sleeper vibration appeared at 670 Hz, and the first-order rigid vibration mode at the frequency of 101 Hz had a significant effect on the condition without the unsupported sleeper. (2) When the sleepers were continuously unsupported, the vibration damping effect of the ballasted bed within the frequency range of 0-450 Hz was better than that at higher frequencies. Within the frequency range of 70-250 Hz, the vibration damping effect of the ballasted bed with unsupported sleepers was better than that without the unsupported sleeper. However, it is not recommended to use unsupported sleepers as a method to reduce the vibration of a ballasted bed in the low frequency range. (3) Owing to the excitation from the wheelset impact, the lateral resistance of the ballasted bed with unsupported sleepers whose hanging height was 30, 60, and 90 mm increased by 37.43%, 12.25%, and 18.23%, respectively, while the lateral resistance of the ballasted bed without the unsupported sleeper remained basically unchanged. (4) The unsupported sleeper could increase the difference in the quality of the ballasted bed between two adjacent sleepers, which could cause differences in the stress conditions between two adjacent sleepers, aggravate the track irregularity, and affect the normal service state of the ballasted bed.   presence of unsupported sleeper may be monitored by using a combination of the lateral resistance of a ballasted bed and the vibration response of the sleeper.