Research Article  Open Access
Jing Zheng, Qingliang Li, Xiaozhen Li, Yunke Luo, "TrainInduced Fluctuating Pressure and Resultant Dynamic Response of Semienclosed Sound Barriers", Shock and Vibration, vol. 2020, Article ID 6901564, 16 pages, 2020. https://doi.org/10.1155/2020/6901564
TrainInduced Fluctuating Pressure and Resultant Dynamic Response of Semienclosed Sound Barriers
Abstract
This paper reports on traininduced fluctuating pressure and consequent dynamic response of semienclosed sound barriers (SESBs), a novel type of sound barriers. A computational fluid dynamic (CFD) model and a finite element (FE) model were established. The in situ vibration measurement of the SESB was introduced for the first time. Pressureinduced vibrations were extracted from the measured results. The analyzing process of the pressureinduced vibration below 20 Hz is verified. Influencing factors of the pressure and vibrations, including train running on the near or farline and the length effect of total top plates, were finally discussed with the dimensionless method. Results show that the distribution of pressure is significantly different, but the vibration laws are similar whether the train is running on the near or farline. The longer is the total length of the top plates, the larger are the pressure and vibration on the inner surface.
1. Introduction
Troubles of environmental noise and vibration triggered by rail transit have attracted a significant amount of research interests recently [1–5]. The wheelrail noise is predominant in overall noise with the train running at a speed lower than 300 km/h [6]. Sound barriers, which have been widely installed along trackside location all over the world, are effective in the noise reduction of rail transit by blocking direct sound propagation [7]. In China, some newstyle sound barriers, such as fully enclosed or semienclosed sound barriers (SESBs), have been applied in trackside engineering to attenuate noise in some demanding regions [8]. Many research studies and exploration have been done on the acoustical performance of conventional and novel sound barriers in the available literature [9–12]; however, the studies on vibration characteristics of sound barriers, especially novel ones, were limited. The violent vibrations of sound barriers with the train speeding up, which may bring about the linked bolts falling off, structural fatigue damage, and displacement of insulation plates, may affect the safe operation of the train, result in structural noise radiation, and even reduce the noise reduction effect of sound barriers. Accordingly, correctly evaluating the vibration response of sound barrier structures is relevant for these technical issues.
If the natural wind impact is not considered, the vibration of sound barriers on highspeed railway bridges arises from wheelrail interaction as well as aerodynamic action related to the pressure fluctuation generated by the transiting train [13]. The analysis methods to predict the wheelrail interaction are often based on the vehicletrack dynamic coupling vibration, taking track roughness as the source of the dynamic force, considering the vehicle as a multibody system composed of carriage body, bogie, primary suspension, secondary suspension, and wheels, solving the dynamic equations of vehicles, rails, and bridges through the finite element model and dynamic theory. Li et al. solved the vibration response of track bridges in time domain by establishing the dynamic equations of vehicles, tracks, and bridges with the Newmarkβ method [14]. Wu and Thompson [15] solved the vibration response of boxgirder bridges in frequency domain by the dynamic compliance method. For recommending the most suitable model in further prediction, Li et al. [16] compared the computation cost and accuracy of results obtained from the mode superposition method in the time domain and power flow models for the vehiclerailbridge coupled system are established in the frequency domain. Zhang and Xia [17] conducted dynamic analysis of the coupled vehiclebridge system based on the intersystem iteration method. These methods all above are often applied in the vibration and vibroacoustic analysis of the coupled system. When sound barriers installed on bridges, these methods are still applicable. Li and Liang [18] investigated the influence of different track structures on the vibration and structureborne noise of elevated concrete box girders with the frequencydomain theoretical model of the vehicletrack coupled system. Song and Li [19] compared the effects of three noise mitigation measures on noise control: installation of a noise barrier, use of a rail pad with a lower stiffness, and adopting of a floating ladder track. With the train speeding up, aerodynamic action and its resultant vibration of sound barrier cannot be ignored. Many related research studies, including basic studies aiming at the qualitative description of the pressure fluctuation, frequencydomain distribution of fluctuating wind, and pressureinduced vibration response, have been carried out. The general rules of fluctuating wind pressure versus time and its related factors were explored through scale measurements, in situ tests, and numerical simulations [20–23]. Consistent conclusions yielded from these studies are that the pressure time history curve includes the head pulse with a positive peak followed by a asymmetric negative peak appearing at the headstock passage, the small amplitude pressure fluctuation caused by the intercarriage gaps, and the tail pulse with a negative and a positive amplitude as a sequence of the tailstock passage. The peak pressure is related to train speed, vehicle type, and distance from sound barrier to the train. Yang et al. [24] decomposed the fluctuating pressure signal, collected in a field test targeting at a sound barrier, in different frequency bands, and figured out that the surface pressure component in frequency band (0–2.5 Hz) is large, while that in higher frequency band (5–10 Hz) is relatively small. Lü et al. [25] figured out that the frequency component of the aerodynamic load is less than 10 Hz. Carassale and Marrè Brunenghi [26] carried out an openair experiment to separate the dynamic response of the aerodynamic forces on a trackside steel frame from the effect of the seismic action caused by the train transit. According to Carassale, the head pulse and the tail pulse on the frequency map have a peak frequency of about 5.8 Hz. Luo and Zhang and Xia [27, 28] derived the calculation formulas for change of wind load acting on the carbody for a l0span simply supported Ushaped girder bridge with 100 m long doubleside 3.5 m barrier, and analyzed the influence of sudden change of wind load on the running safety of the train when a train moves into or out of the wind barrier structure.
Most of mentioned studies are aimed at some other trackside structures or sound barriers with simple structures, such as vertical sound barriers, or simple barriers with low height, whereas for semienclosed sound barriers (SESBs) with complicated structure, the traininduced fluctuating pressure and its resultant dynamic response have not been well studied.
This paper aims to bridge this gap by describing the procedures of a battery of experimental and numerical investigations carried out on a SESB arranged on a highspeed railway bridge. Firstly, traininduced fluctuating pressure and its consequent vibration were simulated numerically, and then, the vibration test of SESB is introduced. The simulated results were compared with the processed measured results for model validation. The verified models were finally applied to carry out parameter studies to investigate and discuss the influences of operating lines and total length of top unit plates on the aerodynamic pressure and dynamic response of the barrier. The results are of major importance and subsequently be used (1) to extract the pressureinduced vibration from original vibration signals according to the frequency component of fluctuating pressure; (2) to obtain the distribution law of fluctuating pressure at different locations of the SESB under different factors; and (3) to derive a fundamental understanding of the vibration response induced by fluctuating pressure of highspeed passing trains.
The composition of the SESB, the CFD model for traininduced fluctuating pressure, and the FE model for pressureinduced vibration were built, respectively, in Section 2. The setup of vibration test and main experimental results were set out in Section 3. Then, in Section 4, the CFD and FE numerical models were verified. Parameter studies were finally carried out in Section 5 to discuss the characteristics of fluctuating pressure and resultant dynamic response of the SESB.
2. Calculating Models
2.1. The Semienclosed Sound Barrier
The target SESB is fixed on the flanges of the 32.6 m long dualtrack simply supported boxgirder viaduct of the Hang ZhouChangsha passenger dedicated line, China. The cross section of the barrier and the bridge girder below it are illustrated in Figure 1. The SESB is 8.15 m in height, 11.7 m in width, and 1.5 km in length, and its main structures with 2 m intervals were steel frames composed of bilateral erect steel studs and top crossbeams. To improve the stability of the steel frames, longitudinal bracings are arranged at different heights along the longitudinal direction. The main noise reduction components of the SESB are the metal sound insulation plates and transparent PC plates installed on the barrier. The steel studs and crossbeams are Hshaped steel with a dimension of 300 mm × 300 mm × 10 mm × 15 mm. Metal plates on the closed side of the SESB, as shown in Figure 2, are 0.140 m thick and 0.45 m or 0.5 m wide, and the PC ones are 0.02 m thick and 1.1 m wide on the side and 1.04 m wide on the top. A total of 14 plates are inserted between the flange of Htype erect steel studs and stuffed rubber into interstices between the plates and the flange of Htype studs. On the top corner of SESB, a curved PC plate with 0.99 m radius was installed. There are 6 PC plates arranged between every two adjacent crossbeams. For further discussion, the track near the enclosed side of barrier is defined as the nearline, and on the contrary, the other one is defined as the farline. Each two adjacent steel columns are regarded as a span of the barrier.
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2.2. CFD Model
2.2.1. Calculation Method
When a highspeed train enters the SESB, wind fluctuating pressure will be generated. The standard twoequation turbulent model and the threedimensional, unsteady, uncompressible RANS equation are solved by FLUENT software to simulate the train wind fluctuation on the inner surface of the SESB. In the process of establishing the CFD model, the grid model of the vehicle and SESB is firstly established (described in detail in Section 2.2.2), and then the size of the calculation domain is determined, in which the dynamic subdomain and the static regions are divided; the boundary conditions are defined.
2.2.2. Models of Train and SESB
The train model used in this paper is China Railways HighSpeed 380B (CRH380B). The headstock and tailstock are 25.7 m long, and every midcarriage is 25 m long. The height and width of CRH380B are 3.89 m and 3.26 m, respectively. The fluctuating pressure wave mainly consists of head pulse and tail pulse induced by the passing headstock and tailstock of the train. Therefore, the aerodynamic behavior of three car models: a lead car, a middle car, and a tail car are very similar with that of eight carriage models or sixteen carriage models. To further simplify the computational model, the train model is properly simplified: neglecting bogies, doors of the train and top pantograph, and in addition, smoothing outer surface of the train as shown in Figure 3(a).
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A CFD model of the SESB was built by establishing the plates with a whole shell which is smoothed. Furthermore, the components such as the track, track slab, ballast, and lower parts of the box girder were neglected in the CFD model. The simplified model of SESB is exhibited in Figure 3(b).
2.2.3. Boundary Conditions
For this model, the size of computational domain is 367.5 m × 168.1 m × 87.7 m as shown in Figure 4. In computational domain, as shown in Figure 5, the vehicle is set to be the dynamic subdomain, the sound barrier is specified to be the static regions, and the contact face is defined as an interface between the two areas. The boundary conditions of the end surfaces where air flows in and out of the computational domain are specified to be the pressure far field. For the computational accuracy and efficiency, the layering mesh method is applied in the following study. In this grid updating method, the computational domain will be divided into smaller subdomains, in which grid reconstruction is easier because of fewer grids, and the static girds at the end of the moving region will be destroyed and created to establish the expected motion of highspeed trains. When the dynamic subdomain moves forward, its front part compresses grids and the rear part stretches the grids. The grid height, 0.5 in dynamic subdomain, decides the time of grids creating and destroying, and the grid destroying coefficient is 0.2 and the grid splitting coefficient is 0.4. The computational domain is generated totally by structured hexahedron grid. A finer grid is used in the area near the surface of the carriage and the SESB. The RANS method of wall function is used to simulate the boundary layer region. The thickness of finest grid on the boundary layer is 0.0012 m. The dimensionless y + for the first node from the walls is 100, and the grid growth factor of boundary layer is 1.1. Except for the cavity of junction and endpoint, quality of the majority exceeds 0.5. The whole model is summed up to 4.62 million nodes and 4.40 million cells. Grid independence test result shows that the average pressure on the monitoring surface obtained by further densifying the grid has little change, and the current model is accurate enough to ensure the validity of the calculated results.
The initial location of highspeed train is 50 m away from the measured section of the barrier. The time step is 0.005 s, and 600 steps in total are calculated. During the progress, the speed of train is set to be 330 km/h on the nearline for the validation of the simulated model and 350 km/h on the near and farline for parameter analysis. The fluctuating pressure on the inner surface of the SESB is explored by the CFD model, and the layout of monitoring faces on the SESB is shown in Figure 6 to capture their average pressure at each time step.
2.3. FE Model for Vibration Analysis
The ABAQUS software was adopted to establish the FE models of vibration responses analysis of SESB under fluctuating pressures. As the fluctuating pressure was loaded on the inner surface and boxgirder bridge was not subjected to this load, concrete bases under the steel studs, instead of the boxgirder bridge, were established as the foundation of SESB and were modelled with eightnode 3D brick elements (C3D8R). The roots of steel studs were fixed on the concrete bases. Steel studs were modelled with fournode shell elements (S4R), as well as the crossbeams and the plates. As the fluctuating pressureinduced vibration is mainly consisted of lowfrequency one, the rubbers stuffed between every two adjacent plates and in the interspaces from plates to steel studs can be omitted because the rubber damping pads can hardly damp vibration of low frequency. The total FE model is shown in Figure 7(a), and its size is consistent with design drawings of SESB. The calculated pressure is expanded from the CFD simulated results of the middle carriage to the pressure of the middle six carriages, they are combined with the head and tail pulse to form the pressure of the eight marshalled train, and they are applied to all the plates in a span on the SESB according to Figure 6. Fluctuating pressure is also applied to the plates on adjacent spans considering the time interval of trains passing through different spans of the SESB. The instantaneous vibration responses such as the dynamic displacement of SESB under pressure can be solved. In Section 4, the calculated results will be compared with the measured ones to verify the applicability of the CFD model and FE model.
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Table 1 tabulates all the material properties of this model. Figures 7(b) and 7(c) present the typical modal vibration shape of SESB, and the first 10 modes of natural frequencies are listed in Table 2. The first mode showing transverse deformation of the SESB is at 4 Hz, and the vertical bending mode is above 5.45 Hz. Whether pressureinduced resonance occurs will be analyzed in Section 3.2.


3. Schemed Field Test
3.1. Testing Instrumentation
A field test was conducted to obtain the vibration response of the target SESB. During the test, three variables were continuously recorded: the train speeds, the location of the moving train, vibration acceleration orthogonal to the test surface by means of a velocimeter, 5 accelerometers and a dynamic data acquisition instrument. Additional information is given below on the location and installation of the monitoring systems.
3.1.1. Accelerometers
The CAYD181 piezoelectric type accelerometers were used to collect the vibration signals. A total of about 5 accelerometers were monitored on the barrier at the midspan of the bridge. Accelerometers were installed through a moving crane as shown in Figures 8(a) and 8(b), which display the distribution and specific locations of accelerometers. V1 and V2 were arranged on the steel stud of the closed side of the SESB at 7.3 m and 1.5 m high from the root, respectively, which were the same height with the top of No.13 plate and No. 3 plate, respectively. V3, V4, and V5 were measuring points on the open side, which were located at 7.3 m, 5.8 m, and 4.45 m high from the root of noise barrier, respectively, the same height with the top of No. 13, No. 10, and No. 7 plate, respectively.
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3.1.2. Train in Field Test
The field test was carried out with a highspeed passing train of CRH380 B. The speed of the train passing through the test section is relatively stable. The traininduced vibration of the SESB is more intensive when the train speed is high enough.
3.1.3. Data Recording System
INV3060S dynamic data acquisition instrument was adopted to collect the vibration information, and the sampling frequency was 25.6 kHz. More than 20 sets of data were monitored on the open side and the closed side, among which 7 sets are about passing train speed around 330 km/h on nearline. Preliminary analysis is carried out to eliminate abnormal data before data processing, and then, statistical average is made on the 7 sets of valid data.
3.2. Testing Results
3.2.1. Measured Results in Time Domain
Now, that the frequency components of fluctuating pressure are different from wheel/rail interaction, the measured vibration signals could be processed by 20 Hz lowpass filtering to obtain lowfrequency vibration induced by the pressure. Figure 9 manifests the filtered results at monitoring points V1–V5 on the SESB in time domain.
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The pressureinduced vibration of all measuring points V1–V5 on the closed side of the SESB is shown in Figures 9(a)–9(d). It indicates that all the waveforms show the similar characteristics with the traininduced pulse. For V2 on the bottom, the vibration response is relatively less than V1, and measured peak accelerations of V1 and V2 are 2.5 m/s^{2} and 0.7 m/s^{2}, respectively. As shown in Figures 9(c)–9(d), the peak accelerations are 2.6 m/s^{2}, 2.2 m/s^{2}, and 1.6 m/s^{2} for V3, V4, and V5, respectively. The vibration on the upper part (V1 and V3) is larger.
3.2.2. Measured Results in Frequency Domain
The measured results from the field test in Figure 9 are transformed into the frequency domain. Because the fluctuating pressure mainly acts within lowfrequency range, Figure 10 only shows the acceleration spectra below 20 Hz. It is indicated that the recorded peak acceleration induced by the pressure is about 0.78 m/s^{2} at 6.8 Hz when the train is on the nearline and is about 0.61 m/s^{2} at 3.3 Hz when the train is on the farline. Compared to the calculated natural frequency exhibited in Table 2, the frequency components of the SESB under the fluctuating pressure are different from the transverse mode at 4 Hz, although many small peaks appear in the acceleration spectra. It can be inferred that the SESB could not resonate with the traininduced pressure. Further investigation should be invested for fatigue issues of the SESB combined with the pressureinduced strains, but these issues are not discussed in this article.
The vibration acceleration from perceptible acceleration (about 10^{−3}m/s^{2}) to intolerable acceleration (about 10^{3}m/s^{2}) for the human body changes up to 1 million times. The vibration acceleration level (VAL) is hence adopted to judge the vibration intensity of the SESB according to Chinese railway industry standard: measurement of railway environmental vibration (TB/T: 3152–2007), and it could be calculated by the following formula:where is vibration acceleration level; is the virtual value of acceleration (m/s^{2}); and is reference acceleration. In addition,where is analysis time and is vibration acceleration at t time.
The vibration is also presented by the way of onethird octave spectra to show the overall trends in the frequency domain, because there are many peaks with different frequency components in the spectra as shown in Figure 10, some of which cannot be reproduced through the simplified numerical calculation, and the vibration trend of each measuring point is not easy to distinguish. Figure 11 illustrates the measured VAL spectra curves of measuring points V1∼V5 from 1∼20 Hz by onethird octave spectra. For V1, V3, V4, and V5 on upper and middle parts of SESB, their vibration characteristics are similar to each other, with dominant frequency range of 2.5 ∼ 5 Hz. This is due to the action of fluctuating wind and the vibration characteristics of sound barrier. While for V2 located at barrier’s lowest part, its vibration peak is relatively smaller and occurred at the frequency band of 10∼20 Hz, where lowfrequency wheel/rail interactions gradually become the main excitations of vibration. However, the pressure action is still dominant within frequency range of 2.5∼5 Hz.
4. Validation Study
Firstly, the CFD numerical method in this paper is verified by the fluctuating wind pressure on the surface of the semicovered snow barrier along the highspeed railway line generated by train traffic in reference [29]. The corresponding model is established by using the same modelling method and parameter values in Section 2.2 of this paper. The comparison between the measured results and the simulation ones at the speed of 252 km/ h is shown in Figure 12. It can be seen that the time history law of the simulated pressure is consistent with the measured one. The “head wave” generates when the head carriage passes through the measurement section, and the “wake wave” generates when the tail carriage passes. The peak pressure of wake wave is about 50%–80% of that of head wave. The peak value of head wave negative pressure and tail wave positive pressure obtained by numerical simulation is slightly larger than the measured value, mainly due to the vehicle model which is Japanese highspeed train, and different from CRH380 B. The simulated results are essentially in agreement with the measured ones, so that the CFD model in this paper can accurately simulate the fluctuating wind pressure caused by the passing trains.
The dynamic displacement from FE results introduced in Section 2.3 is transformed into the dynamic acceleration in time domain by second derivative, and then, the dynamic acceleration in frequency domain is formed by fast Fourier transform. Finally, the VALs in 1/3 octave frequency band are calculated according to formula (1) and (2). To validate the accuracy of numerical models, the processed simulated results are compared with the measured vibration acceleration in time domain and the VALs in frequency domain.
Figure 13 compares the simulated and filtered measured vibration acceleration in time domain of measuring points V1 and V2 on the enclosed side. The simulated accelerations are in accordance with measured ones, and the waveforms of both curves are consistent with the traininduced fluctuating pressure. However, the measured accelerations, especially for the lowest measuring point V2, have some higher frequency components when the intermediate section of a train passing by. This may be both due to the wheelrail interactions the lower parts of SESB suffer more below 20 Hz, and due to the aerodynamic force induced by the intercarriage discontinuities in lowfrequency, which cannot be reproduced by the CFD model. The maximum vibrations of the SESB in lowfrequency component are caused by the head and tail pulse of fluctuating pressure, and the vibration time history obtained by filtering can reflect the dynamic characteristics of the fluctuating wind, which ensure that the 20 Hz lowpass filtering is effective enough in reflecting the pressureinduced vibration.
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Figure 14 exhibits the onethird octave of VAL of measuring points V1 and V2 (enclosed side) and V3 and V5 (open side) derived from the numerical models and filtered measured results. It can be seen that the simulated characteristics of all points are consistent with the measured ones in 1∼10 Hz frequency range, with peak value at 4 Hz, and the measured results are slightly larger than the simulated ones due to the remained lowfrequency wheel/rail forces. In 10∼20 Hz frequency range, the simulated results are not that consistent with the measured ones. This is because the traininduced pressure attenuates in this frequency range and the wheel/rail forces gradually become the main excitations for the vibration of the SESB.
It can also be proved that the vibration accelerations on the open side are consistent with the measured results (V3–V5) in time domain and frequency domain. So far, it can be concluded from above comparison that the fluctuating pressure is the main excitation of vibration in frequency range 1∼20 Hz. Both the calculated acceleration time histories and the VAL spectra of vibration are in accordance with the measured ones and it is not worth bothering to take wheel/rail forces into account when analyzing the characteristics of vibration under pressure. Therefore, the present numerical models are able to reproduce the fluctuating pressure and the resultant dynamic response of the SESB with sufficient accuracy and are can be used for further discussions.
5. Discussion
Based on the calibrated CFD and FE models of aerodynamic pressure and vibration of SESB, the characteristics of aerodynamic pressure and dynamic response under different factors are investigated in this section. Two influential factors, operation line and number of top SIP, are discussed. Train speed is uniformly set as 350 km/h for parameter studies. In order to generalize the results, the response quantities in the following test, including the time and the fluctuating pressure are dimensionless using the train velocity according to the following formulas:where t is the measured time from the passage of the nose of the train with length L, P(T) is the calculated pressure, P_{0} is the undisturbed atmospheric pressure, and ρ is the density of air.
5.1. Effect of Operating Line
The pressure distribution on the inner surface and the vibration law of the steel frame of the SESB are discussed. All the monitoring points are shown in Figure 15, among which O1∼O4, C1∼C4 are on the steel studs of the open and closed sides, and D1∼D5 are on the top crossbeam. The adjacent point intervals on the same side are all 2.5 m.
5.1.1. NearLine
Figure 16 describes the dimensionless pressure and acceleration under dimensionless time when the train is running on the nearline. In this part, the analysis focuses on the head pulse and tail pulse that produce the maximum vibrations of the SESB. It can be seen from Figures 16(a) and 16(b) that the peak pressure coefficient C_{p} of positive pressure of head pulse is the largest at C3–C4 at the bottom of the closed side of the SESB, which is 0.137 at C4 and decreases rapidly in the closed side from C3 to 0.071 at C1. At the top of the SESB, the pressure coefficient is only 1/3 compared to that of C4 on the bottom of the closed side and decreases linearly from 0.062 at D5 to 0.041 at D3. The C_{P} of the negative peak of head pulse, and positive and negative peaks of tail pulse are in the similar cases.
The pressureinduced vibrations at typical location on the steel frame of the SESB are presented in Figures 16(c)–16(e). It is obvious that the large vibration of the SESB is mainly triggered both by the head and tail pulse of the pressure. When each pulse reaches to the first small peak value, the vibrations start to increase; when each pulse reaches to the second peak, the vibration reaches to the first peak. The head pulse excites the vibration of the SESB to arrive at the positive acceleration peak, and the tail pulse causes the negative acceleration peak. The absolute value of the former is greater than that of the latter, because the peak pressure of the head pulse is greater than that of the tail pulse. The vibration always lags the pulses, and the maximum vibration happens with the disappearance of the pulses. Then, the SESB periodically attenuates, and the attenuation period of the vibration on top crossbeam is faster than that of the bilateral steel studs. The vibration is not proportional to the fluctuating pressure, and the largest vibrations on both sides of the SESB appear at C1 and O1, which have the smallest pressure coefficient, and decrease gradually from top part to the bottom part of the steel studs. Due to the connectivity of top crossbeam, the vibration law on the closed side is similar to that on the open side. The greatest vibration of the top crossbeam locates in the middle D3 point and decreases gradually from D3 in the middle to both ends of the crossbeam.
In order to compare the vibration among all monitored points on SESB, Figure 17 exhibits the peak accelerations and the overall VAL on all points. It can be seen that the distribution of accelerations and VAL are basically the same. The peak vibration acceleration is about 2.3 m/s^{2} on both sides and the top of the SESB, which indicates that the sound barrier moves as a whole under the action of fluctuating pressure. As can be seen, for O1 and C1 points on steel studs top, the overall VAL is 115 dB. For measuring points O2∼O4 and C2∼C4, vibrations on the upper points are larger than those of lower ones and vibrations on the closed side are larger than those on the open side. It is because the bottom point is close to the fixed end and is subject to greater constraints, while the upper point is far away from the restraint end. For D1∼D5 on top crossbeam, vibration of measuring point D3 at midspan is the most violent, with the overall VAL of 112 dB. Although much lower pressure loading on the top of SESB, D3 point has smaller constraints but subjected to larger dynamic bending moments. Vibrations of D4 and D5 are slightly larger than D2 and D1, because the former two points are loaded larger pressure than the latter two points.
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5.1.2. FarLine
The dimensionless pressure on the inner surface of the closed side and vibrations during the train running on farline are shown in Figure 18. The peak pressure coefficient as described in Figures 18(a) and 18(b) is 0.033 at C1, which is only 1/4 of the value when the train is running on the nearline, and the pressure distributes almost uniformly when the train is on the farline. Under the action of smaller pressure, the peak vibrations still lag the occurrence of the head and the tail pulse, and the SESB still undergoes periodic attenuation vibration. The vibration periods of both sides are consistent and larger than that of the top crossbeam.
The peak acceleration and overall VAL of the SESB during the farline operation are illustrated in the Figure 19, and it can be seen that the vibration acceleration and overall VAL in farline case are far less than those in the nearline case. Although the vibration on both sides is the same, vibrations on the top are greater while the bottom one is less, but the vibration acceleration and total VAL of each measuring point on the open side are greater than those on the symmetrical position on the closed side. It can be seen that for the measuring points O1 on the top of steel studs on the open side, the peak acceleration is 0.8 m/s^{2} and the overall VAL is 105 dB. Also, for measuring points O2∼O4 and C2∼C4, vibration on the same height are similar to each other and vibrations of upper points are larger than those of lower ones. For the vibration of top crossbeam, the measuring point D3 located at midspan has the largest vibration with 1.1 m/s^{2} peak acceleration and the overall VAL of 108 dB compared to other points on the crossbeam and even the upper measuring C1 and O1 on studs.
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5.2. Effect of Top Plates
The main difference between the SESB and conventional upright sound barrier is whether or not the top of the sound barrier is closed. There are top plates on the SESB. To systematically explore the length effect of top plates on fluctuating pressure and its induced vibration of SESB, seven numerical cases, including sound barriers without any top plates (equivalent to uptight barrier), with only one top corner curve plate, with one top corner curve plate following 1∼5 top plates (1.04 m wide for each plate), are built and discussed under 350 km/h highspeed train operating on the nearline.
5.2.1. Fluctuating Pressure
After calculation, the C_{P} per dimensionless time in different cases is similar to Figures 16(a) and 16(b). According to the calculated results, for all cases with or without the top plates, the greatest fluctuating pressure is still located at the lower part of the steel studs. The peak pressure increases with the extension of the length of total top plates, and the increment in the magnitude of C_{p} is 0.0013/m for the peak positive pressure mainly loading at the lower part (C1–C2) and 0.0014/m for the peak negative pressure appearing at higher position (plates PC2–C3). The pressures on the C1, C2, and C4 points were chosen to be analyzed. Table 3 tabulates the peak positive and negative C_{p} on the three points under each case. As can be seen, with the extension length ranging from 1–3 m, C_{p} enlarges severely. While with the extension length range of 3–5 m, the pressure has small increases.

Without any top plates, the peak pressure loading on C4 is approximately 2.5 times larger than that loading on C1, with peak C_{P} of 0.138 and 0.053, respectively. Compared to pressure of SESB with 6 top plates added to 1 corner curve plate, peak C_{P} on C4 and C1 weakens by about 0.01 and 0.025, respectively, i.e., the reduction rates are 7% and 32%. All the top plates of the SESB mitigate the attenuation rate of fluctuating pressure, which makes the SESB subject to greater fluctuating wind pressure.
In all, the top plates confine the size of air circulating area when calculating the fluctuating pressure, which in turn enlarges the pressure on the inner surface of the barrier, and the pressure acting on the SESB is more likely to be enlarged with the increasing extension length of top plates.
5.2.2. Vibration of SESB
To discuss the influence of the top plates on the vibration of SESB, the vibration distributions of above seven cases are shown in Figure 20. As can be seen, when there are no top plates, the peak vibration acceleration of O1 is 2.3 m/s^{2}, larger than that of C1, about 2.1 m/s^{2}. As for the overall VAL, there are little difference between O1 and C1 because the top crossbeam can effectively connect the open side and closed side of steel studs. Vibrations of the top crossbeam (D1∼D5) are relatively smaller compared to the upper parts of steel studs. This is because there is no aerodynamic pressure loading on the barrier top.
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For the case with one curve plate and one top plate, compared to the case without top plates, the peak acceleration on O1 is 1.5 m/s^{2}, attenuates quickly by 35%. This is because the curve plate and the top plate work as connections, connecting adjacent steel frames to the target steel frame and therefore dissipated the vibration. The overall VAL of O1 and C1 is still consistent for the good connectivity of top cross beam.
As for the cases with different numbers of top plates, the traininduced pressure is magnified with the plate number increasing from 1 to 5, which in turn enlarges the dynamic response of the barrier. The peak acceleration of O1 and C1 per unit extending length of top plates is 0.17 (m/s^{2}) and 0.12 (m/s^{2}), respectively. For the point D3 at the midspan of crossbeam, the increasing rate of peak vibration is 0.17 (m/s^{2})/m. The increase rate of the overall VAL for measuring points O1, C1, and D3 is about 0.5 dB/m, which are consistent with each other.
In all, top plates have significant influences on dynamic response under fluctuating pressure loading on SESB. The pressureinduced vibration at the top of both steel studs and the middle of the crossbeam of SESB is the largest and increases with the increase of the number of top plates. Therefore, in addition to paying attention to the noise reduction characteristics of SESBs, it is also necessary to prevent the engineering problems that may be caused by the highspeed traininduced vibration on the upper and top parts.
6. Conclusions
In this paper, vibration of the SESB is firstly measured in a in situ test and verified by numerical simulation. A series of studies are then carried out on the dimensionless time, and conclusions could be drawn as follows:(1)To distinguish the vibration under the fluctuating pressure from original vibration signals according the frequency component of aerodynamic loads, the effect of wheel/rail forces on vibration responses of the SESB could be neglected when analyzing the characteristics of vibration under aerodynamic pressure.(2)After model calibration through the insite field test, the presented numerical models are able to reproduce the fluctuating pressure and pressureinduced dynamic responses of the SESB with sufficient accuracy.(3)When the train is running on the nearline, the largest pressure coefficient is on the bottom part of the closed side of the SESB and decreases gradually with the height of the steel stud, and pressure on top crossbeam also decreases from the closed side to the open side.(4)Under the action of the head pulse and tail pulse of the fluctuating pressure, the vibration lagging behind will occur and will be periodically attenuated. The vibration laws on both sides of the SESB are the same.(4)The pressureinduced vibration is the largest on the top of the steel studs and in the middle of crossbeam, and the smallest at the bottom of steel studs.(5)Top plates have great influences on the fluctuating pressure and the corresponding vibration. The longer is the total length of the top plates, the greater are the pressure and vibration on the inner surface.
Data Availability
The data used to support the findings of this study are included within the article and available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by the National Natural Science Foundation of China (grant no. 51878565) and the Key Projects of Chongqing Application and Development Plan (no. cstc2014yykfB30003).
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