Acceleration Frequency Characteristics of the Freight-Train-Induced Vibration of the Beijing-Harbin Railway Subgrade

Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nanjing 211189, Jiangsu, China School of Civil Engineering, Southeast University, Nanjing 211189, Jiangsu, China State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Lanzhou 730000, Gansu, China School of Civil Engineering, Qingdao University of Technology, Qingdao 266033, Sandong, China School of Ecological Environment, Hainan Tropical Ocean University, Sanya 572022, Hainan, China


Introduction
A railway subgrade suffers from periodic freezing-thawing cycles, as well as from the train vibration load, and these are the main reasons for roadbed debilitation in seasonally frozen regions. With the development of modern computers and the further application of finite-element numerical calculation methods, research of numerical calculation methods on the subgrade vibration has gradually deepened. Several scholars established a three-dimensional dynamic model to simulate the subgrade structure and track system, predicting the vibration response, and compared the results with those from field monitoring [1][2][3][4]. In model tests, one or more fixed points are applied with harmonic load to simulate the dynamic response and accumulated settlement of high-speed railway roadbeds under the action of train load. To apply moving loads, Momoya et al. [5] built a 1 : 5 indoor model test of ballasting track. Diyaljee [6] and Huang et al. [7] represented the magnitude of the train load and traveling speed through the amplitude and frequency of simple harmonic load. Al Shaer et al. [8] and Brown et al. [9] simulated the moving train load through multiple actuators. In field testing, the Swedish National Railway Authority tested the vibration of the X2000 high-speed train when it was running on soft ground and determined the critical speed in 1998 [10]. Other scholars have carried out field monitoring of several high-speed lines in China, such as the Qin-Shen [11][12][13][14], Beijing-Tianjin [15], Wu-Guang [16][17][18], and He-Ning [19] lines, with the tests covering wheel-rail interaction, dynamic response of the track system, environmental vibration caused by the train load, and accumulated settlement of the subgrade. e results include the influence of train speed on the amplitude-frequency characteristics of roadbed vibration acceleration and the attenuation law of vibration acceleration amplitude and the distribution law of dynamic response and natural frequency. Xia [20] studied the vibration and environmental noise of surrounding buildings caused by traffic load. Ling et al. [21][22][23] conducted field monitoring of the vibration acceleration of a railway subgrade and analyzed the amplitudefrequency characteristics of acceleration in a frozen region.
In summary, the research results that have been published on acceleration monitoring of railway subgrades have mainly concentrated on permafrost regions and high-speed railways, and there are few studies on acceleration monitoring of railway subgrades in seasonally frozen regions. In addition, to the best of our knowledge, no report on the subgrade vibration spectrum caused by heavy trucks exists. In view of this, the acceleration spectrum characteristics caused by heavy freight trains in typical seasonally frozen regions are studied in this paper with the aim of providing the objective basis for railway subgrade design, prevention of debilitation, and disaster assessment in seasonally frozen regions.

Information from Field Experiments. Harbin, located in
Northeast China, is the provincial capital with the highest latitude and lowest temperature in China. e maximum temperature in summer exceeds 37°C, and the minimum temperature in winter is below −35°C.
e maximum freezing depth is approximately 1.8-2 m. e monitoring section is located at k1229 + 095 of Beijing-Harbin railway.
ere are eight measuring points in the monitoring section located at the rail (P1), sleeper (P2), shoulder (P3), embankment (P4), embankment (P5), slope angle (P6), site (P7), and another site (P8). e geometry of the monitoring section, measurement-point layout, and stratigraphic structure are shown in Figure 1. e monitoring instrument adopts an 891-2 vibration measuring instrument, which was developed by the Institute of Engineering Mechanics of the National Seismological Administration. e passband is 0.5-80 Hz, and the acceleration resolution is 1 × 10 −5 . e acquisition equipment applies the INV306 signal acquisition and analysis system developed by China Oriental Vibration and Noise Technology Research Institute.

Introduction of the Monitoring Train.
e vibration acceleration of nine freight trains was monitored. Vehicle parameter information such as vehicle number, vehicle structure, vehicle grouping, driving speed, half of bogie spacing, half of inflexible bogie wheelbase, and sleeper space is shown in Table 1. e schematic of the interaction force between the vehicle and the sleeper is presented in Figure 2

Results and Discussion
e time-frequency characteristics of acceleration in a freezing period were analyzed with a typical freight train as an example. Since point P1 is placed on the track, point P2 is placed on the sleeper and point P3 (the shoulder) on the loose gravel. us, the monitoring results of nine typical freight trains at points P4-P8 were mainly analyzed in this paper. e research results include acceleration-time history, spectrum characteristics, and influencing factors of the freight trains. Figure 3 shows partial monitoring results, including the acceleration-time-history curves of trains 11114, 11138, and 40194 at point P4 and train 11022 at points P4-P7. It can be seen from the graph that the time-history curve of the vehicle is influenced by vehicle formation, load, and speed. e peak and valley of the acceleration-time-history curve of each vehicle appear in pairs. At the point nearest to the center of the rail (P4), the peak time of the acceleration-time-history curve is in clear correspondence to the train wheel. Far from the center of the line (P7), the peak acceleration is small and spindle-shaped. It can be seen that the structural vibration caused by a train passing is transmitted from the track to the subgrade, and the vibration intensity is obviously weakened after passing through sleepers and ballast beds, but its interaction with the wheel and rail still appears correspondingly.

Acceleration Spectrum Analysis.
e internal vibration frequency of railway subgrades is affected by many factors, such as subgrade structure, vehicle type, driving speed, and track irregularity spectrum. Formulas (1)-(3) are used to calculate the vibration frequency caused by the vehicle fixed distance, bogie fixed wheelbase, and sleeper spacing: where v represents the train speed, l c half of vehicle fixed distance, l t half of bogie fixed wheelbase, and l s the sleeper space. e frequency spectrum is obtained by the fast Fourier transform of acceleration-time-history data. Because the acceleration spectrum is discrete, the frequency bands were fitted separately, and the peak fitting method uses the Gaussian function to fit the monitoring data. e abscissa of the center point of the Gaussian function represents the dominant frequency, and the function width represents the bandwidth. In this paper, nine groups of spectrograms for monitoring freight vehicles are given, as shown in Figure 4. Taking freight train 11018 (train formation 2 + 64; traveling speed v � 65 km/h) as an example, it can be seen from        band is f 1, c � 8.3 Hz; the second frequency band is between 25 and 35 Hz, and the corresponding dominant frequency is f 2, c � 33.6 Hz; and the third obvious frequency band is wider, between 35 and 150 Hz, and has the largest vibration energy.
e dominant frequency of the band is f3, c � 60.3 Hz. As can be seen from the graph, each frequency band tends to exhibit normal symmetrical distribution; that is, within each frequency band, the peak frequency is located near the center of the frequency band. e main frequency caused by the vehicle fixed distance, bogie fixed wheelbase, and sleeper space was calculated by formulas (1)   is dominant frequency is not obvious in the spectrum (Figure 4). If the spectrum is amplified locally, it can be found that there is a narrow frequency band near 2 Hz, and the corresponding amplitude is small, which is the reason an in-depth study of the main frequency was not made. f t1,s � 1 × (65/3.6)/(2 × 0.915) � 9.9 Hz, which is similar to the dominant frequency of the first band of 8.3 Hz. f s1,s � 1 × (65/ 3.6)/0.556 � 32.5 Hz, which is similar to the second-band dominant frequency of 33.6 Hz. Figures 4(a)-4(i) show that the freight train has a larger load and more carriages, and the speed of each vehicle is between 60 and 70 km/h. e acceleration frequencies of each train are mainly divided into three bands: the first band's bandwidth is between 0 and 20 Hz, and the dominant frequency is between 8 and 10 Hz; it is mainly caused by the vehicle fixed distance. e second band's bandwidth is between 25 and 35 Hz, and the dominant frequency is near 30 Hz; it is mainly caused by the sleeper space. e bandwidth of the third band is the largest, ranging from 35 to 150 Hz, the dominant frequency is between 50 and 60 Hz, and the vibration energy is the largest. is frequency band is mainly determined by the track irregularity spectrum and field conditions.
Considering that the data of the acceleration spectrogram are discrete and each frequency band basically exhibits normal distribution, the peak values of each frequency band are fitted separately. e frequencies of nine freight trains at different measuring points P4-P8 and different frequency bands were peak-fitted separately, and the corresponding peak frequency was taken as the fitted dominant frequency of the band, drawing the relationship between the fitted dominant frequency (FDF) of each band and the vehicle speed, as shown in Figure 5. It can be seen from the graph that, in the three frequency bands, the linear relationship between the FDF and the vehicle speed has been monotonously increased. Compared with the growth rate of each FDF, it is found that the FDF in the second frequency band clearly increases with increasing driving speed, and the slope is 0.41. e first and third fitting dominant frequencies are not obvious with increasing speed, and the slope is 0.06.

Influence of the Train Formation.
A one-third-octave spectrum is an important means of signal analysis that has the characteristics of wide bandwidth and few spectral lines. To facilitate comparison, the frequency characteristics of acceleration were analyzed by a one-third-octave spectrum.
As can be seen from Figure 6, the dominant frequencies of each vehicle in the first and second bands are different due to the influence of driving speed, while the dominant frequencies in the third band are basically the same. e acceleration amplitudes corresponding to dominant frequencies in each frequency band are clearly different. is is caused by the different number of carriages. Train 11040's acceleration amplitude is the maximum, with a vehicle formation of 2 + 66 and amplitude of approximately 5.8 ms −2 /Hz; train 40194's acceleration amplitude is the smallest, and its vehicle formation is 1 + 31 followed by vehicle 40196, the vehicle formation of which is 1 + 41. It can be seen that vehicle formation has a significant impact on vibration amplitude. Figure 7 shows the onethird-octave spectrum for monitoring the vertical acceleration of passenger and freight trains at P4. Owing to the different sizes of vehicle structures, the frequency spectra of freight and passenger trains are quite different. In the first frequency band, the dominant frequencies of freight and passenger trains are approximately 8 and 10 Hz, respectively; in the second frequency band, the dominant frequencies of freight trains are approximately 30 Hz, while those of passenger trains are not obvious in this frequency band; and in the third band, the dominant frequencies are all approximately 60 Hz.

Influence of the Train Type.
is is because the structure size of freight and passenger trains is different, the number of carriage groups is much larger than that of passenger trains, and the driving speed is less than that of passenger trains. erefore, the dominant frequency in the first and second bands of passenger trains is slightly larger than that of freight trains, but the acceleration amplitude is smaller.

Influence of the Direction.
To analyze the influence of the vibration direction on dominant frequencies in each frequency band, the vibration frequencies of nine trains in the X-, Y-, and Z-directions are compared, as shown in Figure 8. It can be seen from the figure that the dominant frequencies of the X-and Y-directions are basically the same, with obvious differences from the Z-direction. In the first band, the dominant frequencies in the Z-direction are slightly smaller than those in the X-and Y-directions; in the second band, the dominant frequencies in all three directions are basically the same, approximately 30−35 Hz; and in the third band, the dominant frequencies in the Z-direction are near 60 Hz, while the dominant frequencies in the X-and Ydirections are near 100 Hz. e amplitude of acceleration in the Z-direction is obviously larger than that in the X-and Ydirections. is difference is caused by different subgrade structures in different directions. Figure 9 shows the one-third-octave spectrum of freight train 11022 at measuring points P4-P8. It can be seen that the dominant frequencies of each measuring point are basically the same in three bands, i.e., approximately 8-10, 30-35, and 60 Hz. e acceleration amplitude of each measuring point in the three frequency bands decreases with increasing distance from the center of the rail due to the filtering of the site. In the first band, the acceleration amplitude at point P4 is approximately 2.6 ms −2 /Hz, and at point P8, the acceleration amplitude decreases to 1.1 ms −2 /Hz. In the second band, the acceleration amplitude at point P4 is approximately 1.3 ms −2 /Hz, and at point P8, it decreases to 0.4 ms −2 /Hz. In the third band, the acceleration amplitude at point P4 is approximately 4.2 ms −2 /Hz, and the amplitude decreases to 0.4 ms −2 /Hz at point P8. It can be seen that the dominant      Shock and Vibration frequency of the acceleration vibration is almost unchanged, the amplitude attenuation is obvious, and the attenuation rate of the third band is the largest with increasing distance from the orbit.

Conclusions
rough monitoring the acceleration of freight trains running on China's Harbin railway, amplitude-frequency characteristics of vibration acceleration and its influencing factors are described, and the following conclusions are drawn: (1) e acceleration frequencies of railway subgrades caused by freight trains are divided into three main frequency bands. e first frequency band is between 0 and 20 Hz, and the dominant frequency near f t1,s � 1 × v/(2 × l t ); the second frequency band distribution is between 25 and 35 Hz, and the dominant frequency near f s1,s � 1 × v/l s ; and the third frequency band is between 35 and 150 Hz, and the vibration energy is the largest. (2) Because the acceleration spectrum is discrete, the three frequency bands were fitted with peak values separately, and it can be seen that the FDF in the second frequency band clearly increases with increasing driving speed, and the slope is 0.41; the first and third FDFs increase slightly with increasing speed.   structure, and other factors. Among these, train type, traveling speed, and train formation have a significant influence on the dominant frequencies of the first and second bands. Train formation and distance from the track have obvious effects on the amplitude of each frequency band.

Data Availability
Some data used during the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.