Correlation of car-body vibration and train overturning under strong wind conditions

Mechanical Systems and Signal Processing 142 (2020) 106743

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Correlation of car-body vibration and train overturning under strong wind conditions Dongrun Liu a,b,c,d, Gisella Marita Tomasini d, Daniele Rocchi d, Federico Cheli d, Zhaijun Lu a,b,c,⇑, Mu Zhong a,b,c,⇑ a Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, PR China b Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha 410075, PR China c National & Local Joint Engineering Research Centre of Safety Technology for Rail Vehicle, Central South University, Changsha 410075, PR China d Department of Mechanical Engineering, Politecnico di Milano, via La Masa, 1, 20156 Milan, Italy

a r t i c l e

i n f o

Article history: Received 15 October 2019 Received in revised form 13 January 2020 Accepted 15 February 2020

Keywords: Car-body vibration Overturning coefficient Crosswind Full-scale test High-speed train

a b s t r a c t A transient large-amplitude car-body vibration could seriously affect train operational safety and riding comfort when a high-speed train passes through a complex terrain under strong wind conditions. In this study, according to the car-body vibration characteristics under strong wind conditions and the definition of the train overturning coefficient, a railway half-vehicle vibration-overturning analytical model is developed and a formula for the evaluation of the overturning coefficient known the car-body vibration parameters (mainly car-body roll angle lateral displacement) is obtained. Moreover, the formula validity is verified by multi-body nonlinear numerical code simulations and full-scale tests. The results show that the train operational safety under strong wind conditions can be evaluated with a good approximability using the roll angle and the lateral displacement of the center of gravity of the car-body, and the train overturning coefficient calculated using the formula proposed in this paper is well correlated with the overturning coefficient directly measured by instrumented wheelset, and the value amplitude was nearly the same. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction In China, high-speed trains have become the preferred means of transportation. Naturally, it is necessary to continuously improve the operational safety and comfort of high-speed trains. As operational speeds are constantly rising, lightweight body technology has been widely used, and vehicle systems have become much more sensitive to external excitation. This makes it more difficult to ensure operational safety and passenger comfort [1,2]. In particular, additional high-speed railway lines are opening in complex terrain sections in China, bringing a new series of engineering problems. Especially, the wind speed abrupt change along the railway due to the environment winds that are affected by local topography can influence the car-body as a high-speed train passes through, resulting in the sharp swaying of the car-body. This is also known as the ‘‘car swaying” phenomenon [3,4], as shown in Fig. 1. Currently, when the ‘car swaying’ phenomenon occurs during a train’s operation, due to the lack of methods to directly determine the effect of car-body vibration on the operational safety and ride ⇑ Corresponding authors at: Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, PR China. E-mail addresses: [email protected] (Z. Lu), [email protected] (M. Zhong). https://doi.org/10.1016/j.ymssp.2020.106743 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

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D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

Fig. 1. Test results of the car-body roll angle when a train passes through a complex terrain under strong wind speed conditions.

comfort, the driver performs an emergency speed reduction to reduce the swaying of the car-body in accordance with procedures based on individual somatosensory skills. Generally for safety, the train runs at a minimum speed restricted by the line as long as the ‘car swaying’ phenomenon is present, regardless of its amplitude. This can seriously affect not only the efficiency of high-speed trains but also rider experience. According to statistics, in the Baili windy zone of the LanzhouXinjiang high-speed railway in China more than 170 sections have exhibited the ‘car swaying’ phenomenon at an average of one per 0.65 km [5]. Therefore, an investigation on the relationship between car-body transient vibration and train operational safety under strong wind conditions is an urgent engineering problem. In fact, train operational safety and vibration comfort under strong wind conditions differ from those of a train operating in a non-windy environment. When high-speed trains operate under strong wind conditions, especially rapidly changing wind speed conditions, the wind load directly acts on the car-body, resulting in a car-body vibration. Then the force excitation is passed from the car-body to the wheel track through the running gear. This causes a follow-up change in the wheelrail normal force. When a car-body large vibration causes the windward side wheel-rail normal force to be ‘‘0”, the train will reach the critical overturning state (the overturning coefficient is 1.0). Therefore, many scholars have conducted research from the perspective of train aerodynamics. For train operational safety under various terrains (embankments, cuttings, bridges, and windbreaks), the overturning coefficient can be used as a criterion for calculating the critical overturning wind speed of the train, that is, the characteristic wind curve (CWC) [6–9]. CWC represents a useful tool to compare and evaluate the general safety of different trains to crosswind. However, due to the commercial high-speed trains are not able to measure the real-time wind speed that acts on the car-body, this method cannot be used for evaluating the overturning risk during the actual operation of a train, especially in windy harsh environments. Moreover, this CWC uses an ideal wind profile named ‘‘Chinese hat” which represents only one equivalent gust and not a real turbulent wind with spacing scenarios. In addition, the instrumented wheelset that monitors train operational safety is prohibited from being used during normal commercial train operations. This is because, once a normal train wheelset is replaced with an instrumented wheelset, the bogie no longer has any brake capacity. Hence, it is generally used only for special tests [10]. Therefore, the safety status of a train during operation is unknown, and this presents a great challenge to ensuring train operational safety under strong wind conditions. Hence, train overturning accidents caused by strong winds have occurred periodically [11,12]. Therefore, it is not enough to study train operational safety under strong wind conditions only from the perspective of train aerodynamics. It can be seen that the wind load ? car-body vibration ? train overturning is a continuous progressive process. When the external environmental wind speed is small, the car-body small-amplitude vibrations primarily affect train riding comfort. However, when the environmental wind speed is large, especially when wind speed changes, the car-body sways with a large-amplitude. This seriously affects the riding comfort of passengers and may cause the train to overturning due to large-amplitude vibrations of the car-body, as shown in Fig. 2. Therefore, the questions of train operational safety and vibrational comfort under strong wind conditions are coupled. The amplitude of the transient vibration of the car-body is the primary factor that affects train operational safety and comfort. Therefore, studying the relationship between car-body vibration, train overturn safety, and vibrational comfort under strong wind conditions has important engineering significance for ensuring operational safety, ride comfort, and for predicting the train overturning risk. In [13], car-body transient vibration and moment discomfort under strong wind conditions have been studied using a real vehicle test, and a vibration comfort evaluation method, based on the vibration displacement of the car-body has been proposed. Based on the above research, this study further investigates the relationship between car-body vibration and train overturning safety. In particular, starting from a linear analytical model of the railway half-vehicle, the relation between the train overturning coefficient

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

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Fig. 2. Schematic of car-body vibration and train overturning safety under strong wind conditions.

and car-body typical vibration parameters under strong wind conditions is obtained, and then further verified by multi-body dynamic simulations and full-scale experiments. However, until now, there has been little research performed on the car-body vibration and train operational safety under strong wind conditions. It is worth mentioning that the Swedish scholar, Dirk Thomas Matsberg et al. (2008) used a real vehicle test to study the vibration of a car-body relative to the bogie frame when a train passed a curve under strong wind conditions. It was indicated that the vehicle had large displacements in the car-body center of gravity and critical wheelunloading. In addition, the roll angle and lateral displacement of the car-body relative to the bogie frame were linear with the lateral acceleration of the bogie frame [14]. Yu Hibino and Hiroyuki Kannemoto studied the relationship between the carbody lateral vibration and the wheel-unloading ratio using numerical simulation and a real vehicle test. It was found that the maximum lateral acceleration of the car-body was linear with the maximum wheel-unloading ratio [15]. In addition, Jun Xiang studied the train dynamic response under strong wind conditions using a numerical simulation. He found that the effect of a crosswind on the car-body lateral displacement, wheel-unloading ratio, and overturning coefficient was larger than the lateral acceleration, derailment coefficient, and lateral stability index [16]. Dongrun Liu et al. measured car-body vibration displacements when a high-speed train passed through complex terrain under strong wind conditions. It was found that the value of car-body roll angle and the lateral displacement of the center of gravity of the car-body were significantly greater than that of the yaw angle, pitch angle, and vertical displacement of the center of gravity under strong wind conditions, the vibration of the car-body was primarily a low-frequency and large-amplitude rolling and traverse vibration under this conditions. And then the windproof effect of different windbreaks was evaluated based on the car-body roll angle and the lateral displacement of the center of gravity [3,17]. Thomas and Diedrichs investigated the effect of an unsteady aerodynamic force on the dynamic performance of a high-speed train negotiating curves by determining the dynamic responses of a simple artificial gust and more complex gusts. They found that the vehicle reacted with a strong roll response under a simple artificial gust, and the location of the center of gravity of the car-body had a large influence on the crosswind stability of the vehicle [18,19]. It can be seen that it must be better clarify the relationship between the car-body vibration and the train overturning risk. Therefore, this study focuses on this question. In the second part of this work, the simplified analytical model developed is to define the relationship between the overturning coefficient and typical dynamic parameters, which are car-body roll angle and lateral displacement of the center of gravity, and lateral acceleration of the bogie frame and wheelset. In Sections 3 and 4 this model is further verified by multi-body simulation and full-scale experiment test data. 2. Analytical model description and definition 2.1. Car-body vibration characteristics under strong wind conditions To investigate the operational safety of high-speed trains passing through complex terrain sections under strong wind conditions, special tests were conducted by the China Railway Corporation (2014) and the Urumqi Railway Corporation (2016, 2018) [3,5,13,17]. The tests were conducted along the windy zones of the Lanzhou-Xinjiang high-speed railway and the South Xinjiang railway line, as shown in Fig. 3. In these sections, winds can obtain high wind speeds (maximum 64 m/s), long windy periods (some parts suffer winds of level 8 and above for more than 200 days a year), a stable wind direction (almost perpendicular to the railway line), and quickly varying wind speeds. To reduce the effect of winds on train operational safety, different windproof structures were constructed along the railway. These included windbreaks on bridges, embankments, cuttings, and numerous transition structures [3]. During the tests, the car-body vibration parameters were measured using the method given in [5]. The system is able to monitor the roll angle, yaw angle, pitch angle, and lateral and vertical displacement of the car-body with a sampling frequency of 260 Hz, which fully meets the measurement requirements. By considering the aerodynamic performance of the head car in the worst-case scenario under strong wind conditions and the requirements of the field measurement and the actual structure of the car-body, the NO.1 car was selected as the test car. In the test, the first axle of the first bogie

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Fig. 3. The topographies of the test railway lines (the Lanzhou-Xinjiang high-speed railway line and the South Xinjiang railway line).

of the head car replaced the instrumented wheelset. The arrangement of the equipment and installation of the measurement system are as shown in Fig. 4. Fig. 5 shows the test result of the NO.1 car-body vibration displacements when the test train was traveling at 60 km/h (CRH5G, 2016). It can be seen that the car-body roll angle and the lateral displacement of the gravity center of the car body were significantly greater than the value of the yaw angle, pitch angle, and vertical displacement of the gravity center of the car-body. In most sections, the roll angle and the lateral displacement of the gravity center significantly fluctuated, and at some moments, there was a significant abrupt change. The value of the yaw angle and the vertical displacement also changed correspondingly, but the amplitude change was small. Fig. 6 shows the fast Fourier transform (FFT) analysis result of the carbody roll angle and the lateral displacement of the gravity center. It can be seen that the rolling and transverse vibration of the car-body were mainly distributed in the low-frequency band below 1 Hz, and its primary frequency is 0.59 Hz (CRH5G). Therefore, it can be concluded that the car-body vibration under strong wind conditions is primarily a low-frequency and large-amplitude rolling and transverse motion, and the car-body rolling motion and transverse motion are coupled. 2.2. Car-body vibration-overturning analytical model As mentioned above, when the windward side wheel-rail normal force is equal to zero, the vehicle will reach the critical overturning state. Therefore, the train overturning coefficient is the most important indicator for assessing train operational safety under strong wind conditions. Therefore, to study the relationship between the car-body vibration and the train overturning safety, a vehicle vibration-overturning analytical model was developed, as shown in Fig. 7. According to the Chinese standard (GB-5599-85) [20], the definition of the overturning coefficient is:

D¼

PL  PW PL  PW ¼ < 0:8 PL þ PW P st

ð1Þ

where PL is the wheel-rail vertical force on the leeward side; P W is the wheel-rail vertical force on the windward side; and Pst is the total static vertical force of the wheel-set. It can be seen from the definition that the key factor in obtaining the overturning coefficient is the solution of the difference between the wheel and rail vertical force of the windward and leeward sides, PL  PW . Therefore, the equations of motion of the wheelset in the analytical model were written.

F PyL þ F PyW þ mWs  ayWs ¼ Q L þ Q W

ð2Þ

Fig. 4. Test equipment installation.

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

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Fig. 5. Test results of car-body vibration displacement at 60 km/h (CRH-5G): (a) roll angle, yaw angle, and pitch angle; and (b) lateral and vertical displacement of gravity center of car-body [13].

Fig. 6. FFT spectra of car-body vibration displacement during high-speed train operation at 60 km/h: (a) car-body roll angle, (b) lateral displacement of gravity center of car body.

F PzL þ F PzW þ GWs þ mWs  azWs ¼ PL þ PW

ð3Þ

F PzL  bp þ Q L  r þ Q W  r þ PW  ðb þ yÞ ¼ F PzW  bs þ PL  ðb - yÞ

ð4Þ

PL  PW ¼ þ D¼

ðF PzL F PzW Þbp þðF PyL þF PyW þmWs ayWs Þr b ðF PzL þF PzW þGWs þmWs azWs Þy b

ðF PzL  F PzW Þ  bp þ ðF PyL þ F PyW þ mWs  ayWs Þ  r y þ b  Pst b

ð5Þ

ð6Þ

where F PyL and F PyW are the leeward side and windward side lateral forces of the primary suspension system acting on the wheelset, respectively; F PzL and F PzW are the leeward side and windward side vertical forces of the primary suspension system acting on the wheelset, respectively; mWs is the mass of the wheelset; and ayWs is the lateral acceleration of the wheelset. azWs is the vertical acceleration of the wheelset; Q L and Q W are the leeward side and windward side wheel-rail lateral forces, respectively; PL and PW are the leeward side and windward side wheel-rail vertical forces, respectively; GWs is the

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Fig. 7. Analytical model of the car-body vibration and train overturning.

gravitational force of the wheelset; y is the displacement of the contact point on the wheel; and r is the rolling radius of the wheelset. It can be seen from Eq. (6) that the train overturning coefficient is composed of four parts: the vertical force difference of the primary suspension system on the windward and the leeward sides, the lateral force of the primary suspension system, the lateral inertial force of the wheelset, and the lateral displacement of the wheel-rail contact point. When a high-speed train operates under strong wind conditions, the environmental wind load directly acts on the carbody as an input. This results in large-amplitude vibrations of the car-body, causing the air spring system and the primary suspension system on the windward and the leeward sides to be continuously compressed and stretched. Then the force excitation is passed from the car-body to the wheel and track, causing a follow-up change in the wheel-rail normal force. Therefore, assuming that the moment, M RC (the reference point is the position of the car-body’s center of gravity), under strong wind conditions causes the roll angle of the car-body relative to the rail to be hC , the roll angle of the car-body relative to the bogie frame to be hB , and the roll angle of the bogie frame relative to the rail to be hWs . Hence,

hC ¼ hB þ hWs

ð7Þ

Moreover, since the car-body and the bogie frame are primarily connected by the secondary air spring system, the roll angle, hB , of the car-body relative to the bogie frame can be obtained using the difference in the vertical dynamic deflection of the secondary air spring.

hB ¼

DlSW  DlSL 2bS

ð8Þ

where DlSW and DlSL are the dynamic deflections on the windward and leeward sides of the secondary air spring suspension, respectively, with 2bS spacing between them. Similarly, the roll angle, hWs , of the bogie frame relative to the wheelset can be obtained using the difference in the vertical dynamic deflection of the primary suspension system:

hWs ¼

DlPW  DlPL 2bP

ð9Þ

where DlPW and DlPL are dynamic deflections on the windward and leeward sides of the primary spring suspension, respectively, with 2bP spacing between them. Similarly, assuming that the lateral force, F TC , the strong wind conditions causes the lateral displacement of the car-body’s center of gravity relative to the rail to be yC , the lateral displacement of the car-body’s center of gravity relative to the bogie frame to be yB , and the lateral displacement of bogie frame relative to the wheelset to be yWs . Hence,

yC ¼ yB þ yWs

ð10Þ

yB ¼ DySW ¼ DySL

ð11Þ

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

yWs ¼ DyPW ¼ DyPL

7

ð12Þ

where DySW and DySL are the lateral displacement of the car-body relative to the bogie frame on the windward and leeward sides, respectively; and DyPW and DyPL are the lateral displacement of the bogie frame relative to the wheelset on the windward and leeward sides, respectively. In addition, when the train is in operation, the vertical force difference and the lateral force of the suspension system can be calculated using the vertical and lateral displacement caused by the displacement of the suspension system on both sides (assuming linear elastic behavior). Because the vibration of the car-body under strong wind conditions is a low-frequency vibration below 1 Hz, the influence of damping was not considered in this model [21]. Hence,

F PzL  F PzW ¼ K Pv  ðDlPW  DlPL Þ

ð13Þ

F PyL þ F PyW ¼ K Pt  ðDyPW þ DyPL Þ

ð14Þ

F SzL  F SzW ¼ K Sv  ðDlSW  DlSL Þ

ð15Þ

F SyL þ F SyW ¼ K St  ðDySW þ DySL Þ

ð16Þ

By substituting Eqs. (9) (12)–(14) into Eq. (6): 2

D ¼ hWs  2

K P v  bp K Pt  r mWs  ayWs  r y þ yWs  2 þ þ b  Pst b  Pst b b  Pst

ð17Þ

where K Pt and K Pv are the lateral stiffness and vertical stiffness of the primary suspension system, respectively. In addition, according to the European standard EN14067-6 [8], the maximum displacement of the wheel contact point in the flange direction is 0.028 m with respect to the track gauge of 1.435 m; hence the wheel-rail contact point displacement during train operation is very small. Considering the maximum displacement of the contact point, the impact of the wheelrail contact point offset on the train overturning coefficient is less than 4.7%. The current technology cannot accurately measure the wheel-rail contact point displacement in real-time during train operation. Therefore, the influence of the wheel-rail contact point displacement on the measurement results is not considered in this paper. Hence, the equation is corrected using a constant coefficient, j. Hence, 2

D ¼ hWs  2

K P v  bp K Pt  r mWs  ayWs  r þ yWs  2 þ þj b  Pst b  Pst b  Pst

ð18Þ

It can be seen that the train overturning coefficient is positively correlated with the roll angle and the lateral displacement of the bogie frame relative to the wheelset and the lateral acceleration of the wheelset. Further analysis was conducted on the system consisting of a car-body and bogie frame (the reference point was the position of the car-body’s center of gravity). According to the research [21], the train overturning safety cannot consider the vertical vibration inertia of the train sprung mass. Therefore, this study did not consider the influence of the vertical inertia force of the bogie frame and did not consider the influence of the spring offset on the stiffness. In addition, when a train is fully loaded with passengers, it will cause the car-body’s center of gravity to move down [22]. Because this study considers the worst-case scenario, it only considers the situation when a train has no passengers and no load. In addition, the vibration of the car-body under strong wind conditions consists of lower-center rolling motion, so the car-body vertical displacement caused by the rolling motion is not considered. Hence, the lateral force and moment equation of the car-body with respect to the center of gravity can be written.

MRC þ ðF 0SyW þ F 0SyL Þ  hC þ F 0SZW  bS  F 0SZL  bS ¼ 0

ð19Þ

MRC þ 2ðF 0PyW þ F 0PyL ÞhCB þ 2F 0PZW ðbP þ yB Þ  2F 0PZL ðbP  yB Þ  mB  ayB  hCB  mB  g  yB ¼ 0

ð20Þ

F TC þ mC  ayC ¼ F 0SyW þ F 0SyL

ð21Þ

F TC þ mC  ayC ¼ 2ðF 0PyW þ F 0PyL Þ  mB  ayB

ð22Þ

By combining Eqs. (7)–(16), the following is obtained:

yWs ¼ yC 

K St mB þ ayB  2K Pt þ K St 4K Pt þ 2K St 2

hWs ¼ hC 

K Sv b S 2K Pv 

2 bP

2

þ K Sv bS



ð23Þ

A  yC þ B  ayB 2

2

4K Pv  bP þ 2K Sv bS

ð24Þ

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D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

where F SyL and F SyW are the leeward side and windward side lateral forces of the secondary air spring suspension acting on the bogie frame, respectively; F SzL and F SzW are the leeward side and windward side vertical forces of the secondary air spring suspension acting on the bogie frame, respectively. mC is the mass of the half car-body; mB is the mass of the bogie frame; and ayC is the lateral acceleration of the car-body. ayB is the lateral acceleration of the bogie frame; g is the gravitational acceleration; hCB is the height of the center of gravity of the car-body from the center of gravity of the bogie frame; hC is the height of the center of gravity of the car-body from the bottom of the car-body; 2bS is the distance between the secondary air spring suspension; and 2bP is the distance between the primary spring suspension.

A¼

2mB gK Pt 4K St K Pt ðhC  hCB Þ þ 2K Pt þ K St 2K Pt þ K St m2B gK Pt

B¼

2K Pt K St þ

K 2St

þ

2mB K Pt ðhC  hCB Þ m2  g þ mB ðhCB  hC Þ  B 2K Pt þ K St 2K St

Substituting Eqs. (23) and (24) into Eq. (18) yields:

D ¼ C  hC þ E  yC þ H  ayB þ I  ayWs þ j

ð25Þ

where 2 2

C¼

2K Pv K Sv bS bp ð2K Pv 

2 bP

2

þ K Sv bS Þb  Pst 2

E¼

AK Pv  bp 2K St K Pt  r  ð2K Pt þ K St Þb  P st 2K Pv  b2P þ K Sv b2S

H¼

B  K P v  bp mB K Pt  r  ð2K Pt þ K St Þb  P st ð2K Pv  b2P þ K Sv b2S Þb  Pst

2

I¼

mWs  r : b  Pst

It can be seen from Eq. (25) that the train overturning coefficient is positively correlated with the roll angle, the lateral displacement of the car-body’s center of gravity, and the lateral acceleration of the bogie frame and wheelset. 3. The study by multi-body simulation The relation in Eq. (25) proposed in the previous section can be verified by multi-body approach and numerically simulating dynamic behavior of a rail-vehicle under strong wind conditions. The simulation process of the crosswind-vehicle model is shown in Fig. 8. 3.1. Definition of the crosswind-vehicle model 3.1.1. Crosswind model In [23,24], the wind speed along a railway was measured using the onboard anemometer be mounted on the high-speed train when passed through complex terrain under strong wind conditions, as shown in Fig. 9(a). It can be seen that the environmental wind was affected by the terrain along the railway line, and the wind speed curve shows a significant linear abrupt change at some instants, while the wind speed remained constant during other periods. For the train, when the train passed a wind speed abrupt change section at a higher speed, a continuous wind speed abrupt change occurred. Therefore, considering the rate of wind speed change, the duration of peak-wind, the amplitude of wind speed change, and the interval time between two adjacent wind speed changes, the crosswind model shown in the Fig. 9(b) was established (note that this investigation studies the only scenario where the wind angle is 90°). In this crosswind model, the parameter definitions are as follows: Dt si is ramp time; v wi is peak wind speed and at different mean wind speeds v w;mean ; DtLi is peak wind-speed duration; and Dt g is interval time.

Fig. 8. Schematic of the crosswind-vehicle model.

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35

vw1, max t2 t3 Wind speed (m/s)

Wind speed (m/s)

30

25

20

15

—— Wind speed —— Cross-wind profile

10

19:04:00

19:06:00

vw2, max t6 t7

vw2

vw1

t0 vw, mean

t1 ts1

t5

t4 tL1 ts1

tg

ts2

t8

t9

tL2 ts2

19:08:00

Time (s)

Time (s)

Fig. 9. (a) Wind speed curve measured by the on-board anemometer when the train passed through complex terrain sections under strong winds [4], and (b) profile of the wind speed model.

Fig. 10. Definition of the composed wind and the yaw angle.

3.1.2. Train aerodynamic forces and the dynamic model 3.1.2.1. Train aerodynamic forces. It can be seen from the above discussion that the crosswind model involves many parameters. It is very uneconomical to use an unsteady calculation method to calculate the aerodynamic load when the different parameters change [25,26], which requires a large number of computing resources and a long duration. Therefore, some scholars have proposed to calculate the aerodynamic load using quasi-static methods [27–29]. Because this study focuses on the correlation between car-body vibration and train overturning safety under strong wind conditions, the aerodynamic load was calculated using quasi-static methods. The calculation process is as follows: (1) Calculate (CFD simulation) or measure (wind tunnel test) the aerodynamic force coefficients under different wind and running speeds to obtain the aerodynamic force coefficients under different yaw angles. (2) Fitting the relationship between the aerodynamic coefficients and the yaw angle, and obtain the fitting polynomial between the yaw angle and the different aerodynamic coefficients. (3) According to the crosswind speed curve, the aerodynamic force on the train under arbitrary conditions can be evaluated using Eqs. (26)–(29). (4) Loading the obtained pneumatic time curve on the dynamic model.

Fi ¼

1 qSC C Fi ðbÞv 2res 2

Mi ¼

1 qSC LC C Mi ðbÞv 2res 2

v res ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðv t þ v w sinaÞ þ ðv w cosaÞ2

b ¼ arctan

v w cosa v t þ v w sina

ð26Þ ð27Þ ð28Þ ð29Þ

Where the relationship of the composed wind and the yaw angle as shown in Fig. 10. In [4], the accuracy of the quasi-static calculation method has been validated in detail. The quasi-static results and the unsteady results are similar, and the variation trends are the same. However, a ‘time-lag’ exists between the quasi-static results and the unsteady results because of the quasi-static method does not consider the response time of the wind and the aerodynamic force. However, this study focuses on the correlation of car-body vibration and the train overturning

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D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

coefficient; therefore, the ‘time-lag’ will not affect the results in this study. In addition, the effects of the pitch moment and yaw moments on train overturning under strong winds are small [8]; therefore, this study does not consider the effects abrupt increases in the pitch and yaw moments at the inflection point of the wind curve. Therefore, the approximation introduced by quasi-steady approach will not significantly affect the results in this study. In this study, the aerodynamic force coefficients were obtained from the CFD simulation. The CFD simulation set up and verification of the numerical calculation method has been described in detail in [4]. Fig. 11 shows the CFD result of the head car force coefficient of the wind tunnel test results [30] and a comparison with the numerical calculation results. It can be seen that the simulation results agree well with the test results, and the maximum error is within 10%, which meets the requirements of this study. 3.1.2.2. Dynamic model of the high-speed train. The most critical case was used in this study where a Chinese CRH2 high-speed train trailer was considered as the research object [8]. In addition, the software SIMPACK was used for the analysis [31]. Fig. 12 shows a side view schematic of a CRH2 vehicle trailer, and the basic data have already been described in detail in [4]. The vehicle model contains a car-body, two bogies, and four pairs of wheelsets. In this study, these subsystems are regarded as rigid bodies that are connected by joints or constraints. The primary suspension between the wheelset and bogie frame contains a helical spring, a vertical damper, and an axle-box. The second suspension between the bogie frame and the car-body contains two air springs, one lateral damper, one longitudinal damper (anti-yaw motion damper), and one lateral bump stop. The track model is defined as inertia fixed rails. According to the standard [8], the effect of track irregularity was not considered in this study. In addition, only a train running on a straight track was considered in this study. The aerodynamic loads acted on the aerodynamic reference point (the center of gravity of the car-body projected onto the horizontal plane passing the top of the rails). The accuracy of the dynamic model was critical for this study. In this study, the dynamic model was verified by comparing the simulated car-body roll angle with the full-scale test result under the same conditions. The test result was obtained from [3]. The car-body roll angle was measured when the train was passing through the embankment-rectangular transitioncutting section (K3033 + 194) in the windy zone of the Lanzhou-Xinjiang high-speed railway in China. The aerodynamic loads for the dynamic model validation were obtained from [32]. The [32] studied the aerodynamic characteristics when the train passed through the same section (K3033 + 194) by using CFD simulations (the terrain model was simplified appropriately). Fig. 13 shows the car-body roll angle comparison results obtained from the simulation and test. It can be seen that the calculation and test result showed a good follow-up characteristic, and the maximum error is small. Hence, the proposed dynamic model meets the requirements of this study. 3.2. Calculation results and discussion 3.2.1. Dynamic response of the high-speed train subjected to different wind speed time-histories In this section, the wind speed time-history was derived from the crosswind profile shown in Fig. 6(b), and the case of two adjacent wind speed changes in the same direction (Case A) and in the reverse direction (Case B) were considered, as shown in Table 1. Fig. 14 shows the train safety index overturning coefficient, derailment coefficient (Additionally, wind speed abrupt change not only affect the train overturning safety but also the derailment safety. To further clarify the risk relationship

2.5

2.5

Wind tunnel test Numerical simulation Error

2.0

2.0

1.5

CFz

CFy

1.5

1.0

1.0

0.5

0.5

0.0

Wind tunnel test Numerical simulation Error

6% 3%

2%

0

5

5% 4%

10

Yaw angle (°)

15

6%

0.0

5%

5%

4%

10

15

20

2%

20

0

5

Yaw angle (°)

Fig. 11. Comparison between the head-car force coefficient values obtained using the wind tunnel test and the CFD: (a) Side force, (b) Lift force.

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

11

Fig. 12. Schematic side view of a CRH2 vehicle trailer.

between train overturning and derailment, this section also considers the train derailment coefficient, defined as Eq. (30)), car-body and bogie frame roll angle, bogie frame and wheelset lateral acceleration, secondary air spring, and primary suspension system vertical displacement time history curve under the case A. It can be seen that the train overturning coefficient and the roll angle and lateral displacement of the center of gravity of the car-body are consistent with extended time. This indicates that there is a strong linear relationship between the rolling and traverse vibration of the car-body and the overturning of the train. It can also be seen from Fig. 14 that the roll angle and the lateral displacement of the center of gravity of the car-body, the roll angle of the bogie frame, the vertical displacement of the secondary air spring, and the primary suspension system time histories also show good synchronism. This explains that under strong wind conditions, the transmission path of the load acting on the vehicle system is transmitted from the car-body to the wheel-rail, and the influence of the environmental wind on train safety is much greater than other factors. In addition, It can be seen that when the train overturning coefficient is close to the critical value of 0.8 [20], the derailment coefficient did not reach the critical value of 0.8 [20], and when the overturning coefficient reached the maximum value, the derailment coefficient value was small. This indicated that the overturning coefficient was more likely to exceed the critical value of the derailment coefficient under strong wind conditions, hence, it is reasonable to use the overturning coefficient to evaluate train operational safety under these conditions. In addition, it can also be seen that the train derailment coefficient reaches the extreme value at 8.34 s and 9.82 s due to the abrupt change in wind speed. Also, the lateral acceleration of the bogie frame and the wheelset also reach the maximum value. However, when the train overturning coefficient reaches the maximum value, the lateral acceleration values of the bogie frame and the wheelset are small at that time. This indicates that the lateral accelerations of the bogie frame and the wheelset have a greater influence on the derailment coefficient than the train overturning coefficient.

T¼

Q < 0:8 P

ð30Þ

where P is the wheel-rail vertical force, Q is the wheel-rail lateral force. Fig. 15 shows the vibration response time histories of the car-body, frame bogie, and wheelset under the case B. It can be seen that the changing trend is basically the same as that when the wind speed changed in the same direction, and this analysis will not be repeated here. However, it is worth noting that when the wind speed direction reverses, the vertical displacements of the secondary air spring and the car-body’s center of gravity appear as the anti-phase vibration to the roll angle and the lateral displacement of the center of gravity of the car-body after one or two periods, and then they resume to the same phase vibration. This is due to the fact that initial direction of the wind load was opposite to the direction of the car-body motion when the wind direction suddenly reversed. Further research is needed regarding this aspect of motion.

1.5

Roll angle (°)

1.0

0.5

0.0

-0.5

-1.0 66

—— Experient result —— Simulation result 69

72

75

78

Time (s) Fig. 13. Comparison of the roll angle obtained from the simulation and from the field test.

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D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

Table 1 Simulation cases. Case

Train speed (km/h)

v w;mean

A B

250

15

v wi;max

(m/s)

Lateral acceleration (m/s2)

v w1 ¼ v w2 ¼ 30 v w1 ¼ v w2 ¼ 25

0.4

Safety index

(m/s)

Overturning coefficient Derailment coefficient

0.0 -0.4 -0.8 8.34 s

-1.2

9.82 s

Dt Li (s)

Dtg (s)

Dts1 ¼ Dt s2 ¼ 0:3

Dt L1 ¼ DtL2 ¼ 0:3

0.3

9.82 s

8.34 s

Bogie frame Wheelset

4 2 0 -2 -4

0.05

Displacement (m)

1.5

Roll angle (°)

6

Dtsi (s)

0.0 -1.5

Bogie frame Car-body

-3.0 8

9

10

11

12

13

14

0.00 -0.05 -0.10 -0.15 -0.20

15

Second air spring vertical Primary spring vertical Car-body lateral 8

9

10

11

12

13

14

15

Time (s)

Time (s)

Fig. 14. Times histories of the different vibration parameters and train safety index under the case A: (a) train overturning coefficient and derailment coefficient and the roll angle of car-body and bogie frame; (b) the lateral acceleration of the bogie frame and wheelset and the vertical displacement of secondary air spring, primary suspension system, and the lateral displacement of the center of gravity of the car-body.

Lateral acceleration (m/s2)

Safety index

1.2

Overturning coefficient Derailment coefficient

0.8 0.4 0.0 -0.4

9.8s

-0.8

10.76s

-1.2

Bogie frame Car-body

1.5

Displacement (m)

Roll angle (°)

3.0

0.0 -1.5 -3.0 8

9

10

11

12

Time (s)

13

14

15

4

9.8s

Bogie frame Wheelset

10.76s

2 0 -2 -4 -6

0.12

Second air spring vertical Primary spring vertical Car-body lateral

0.06 0.00 -0.06 -0.12

Wind speed direction reverses 8

9

10

11

12

13

14

15

Time (s)

Fig. 15. Time histories of the different vibration parameters and train safety indexes under the case B: (a) train overturning coefficient and derailment coefficient and the roll angle of the car-body and bogie frame; (b) the lateral acceleration of the bogie frame and wheelset and the vertical displacement of the secondary air spring, primary suspension system, and the lateral displacement of the center of gravity of the car-body.

To further investigate the influence of car-body vibration on train overturning under strong wind conditions, a fast Fourier transform (FFT) analysis was conducted on the train overturning coefficient, the roll angle, the lateral displacement of the car-body’s center of gravity, and the lateral acceleration of the bogie frame and wheelset, as shown in Figs. 16 and 17. The results show that the vibration primary frequency of the train overturning coefficient was 0.84 Hz. Similarly, the vibration primary frequencies of the roll angle and lateral displacement were also 0.84 Hz, the vibration primary frequencies of the bogie frame and the wheelset lateral acceleration were higher than 1 Hz. Therefore, it can be concluded that the train

13

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743 0.4

0.10

0.84 Hz

0.06

0.04

0.2

0.1

0.02

0.00

0.84 Hz

0.3

Amplitude

Amplitude

0.08

Car body roll angle Car body lateral diaplacement Bogie frame lateral acceleration Wheelset lateral acceleration

2

4

6

8

0.0

10

0

4

Frequency (Hz)

8

12

16

20

Frequency (Hz)

Fig. 16. FFT spectra of the train overturning coefficient and different vibration parameters under the case A. (a) Train overturning coefficient; and (b) the roll angle and lateral displacement of the center of gravity of the car-body and the lateral acceleration of the bogie frame and wheelset.

0.4

0.10

0.84 Hz

Car body roll angle Car body lateral diaplacement Bogie frame lateral acceleration Wheelset lateral acceleration

0.84 Hz

0.3

0.06

Amplitude

Amplitude

0.08

0.04

0.1

0.02

0.00

0.2

2

4

6

Frequency (Hz)

8

10

0.0

0

4

8

12

16

20

Frequency (Hz)

Fig. 17. FFT spectra of the train overturning coefficient and different vibration parameters under the case B. (a) Train overturning coefficient; and (b) the roll angle and lateral displacement of the center of gravity of the car-body and the lateral acceleration of the bogie frame and wheelset.

overturning coefficient vibration primary frequency was close to the train natural frequency of the car-body lower-center rolling motion under strong wind conditions. In addition, the vibrations that may cause the train to overturn were primarily the car-body lower-center rolling and traverse motions. The effect of the rolling and traversing motions of the car-body on train overturning under strong wind conditions was much greater than the lateral acceleration of the bogie frame and wheelset. Therefore, when analyzing the influence of the car-body vibration on train overturning, 1 Hz low-pass filtering should be applied to the lateral acceleration of the bogie frame and the wheelset. 3.2.2. Influence of considering different vibration parameters on train overturning coefficient under different cases. According to the previous analysis, the influence of the rolling and traverse motions of the car-body on the overturning of the train was much greater than the lateral acceleration of the bogie frame and wheelset. To further analyze the influence of the bogie frame and wheelset lateral acceleration on the overturning coefficient of the train, the overturning coefficient, Di , calculated using Eq. (25) was compared with the train overturning coefficient, D, calculated directly by the wheel-rail force in SIMPACK. This section does not consider the influence of the correction factor, j, and it compares the result under different cases: (1) Considering the roll angle, the lateral displacement of the center of gravity of the car-body, and the lateral acceleration of the bogie frame and wheelset, D1 . (2) Considering the roll angle, the lateral displacement of the center of gravity of

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D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

the car-body, and the lateral acceleration of the bogie frame, D2 . (3) Only considering the roll angle and the lateral displacement of the center of gravity of the car-body, D3 . These cases are shown in Table 2: where the coefficients C, E, H, and I are calculated by substituting the basic parameters of the vehicle. In this simulation, C ¼ 0:1694(the roll angle unit is °), E ¼ 2:45(the lateral displacement unit is m), H ¼ 0:067(the acceleration unit is m/s2), and I ¼ 0:0297(the acceleration unit is m/s2). In addition, 1 Hz low-pass filtering was applied to the lateral acceleration. Fig. 18 shows the calculation results of the overturning coefficient under different cases. It can be seen that, overall, in the results obtained from Di is consistent with the changing trend of the train overturning coefficient, D, calculated by the wheelrail force. This result indicates that it is possible to predict train overturning safety by the vibration parameters of the carbody. When the overturning coefficient value is larger, D1 is closer to D than D2 and D3 , but the difference between D1 and D2 , D3 is small. When the overturning coefficient value is small, the values of D1 , D2 , D3 are basically the same with D, which indicates that the wheel-rail contact point offset on the train overturning coefficient is negligible under this condition. In addition, it can be seen from the figure that the wind speed reverses direction, the error between D and Di is significant, however, the overturning coefficient value at the moment of the wind speed reverses direction is small, the overall impact on train overturning is not significant, hence, the error of the moment of the wind speed reverses direction can be ignored. To further investigate the accuracy of predict train overturning safety by the car-body vibration parameters, the error coefficient (CerrD) be defined as described in Eq. (31). Considering that during the train operation under strong wind conditions usually focus on the maximum value of the overturning coefficient, the values of CerrD at the moment of P1, P2, and P3 were calculated, as shown in Fig. 19. It can be seen that, overall, the value of CerrD1 is around 0.04–0.1, CerrD2 is around 0.03– 0.12, CerrD3 is around 0.04–0.12, the CerrD1 is small than CerrD2 and CerrD3, the CerrD2 and CerrD3 are close. This means that evaluating the overturning coefficient by D1 , D2 , D3 is feasible in engineering, the lateral acceleration of the bogie frame and wheelset have an impact on train overturning, and the effect of the bogie frame lateral acceleration on train overturning is larger than wheelset lateral acceleration, however, the overall impact is not significant, and general engineering applications can ignore the effects of the bogie frame and wheelset lateral acceleration that can be corrected by a coefficient, as shown in Eq. (32).

  Di  D  C errDi ¼  D 

ð31Þ

D ¼ C  hC þ E  yC þ j

ð32Þ

4. The study by full-scale test In the Lanzhou-Xinjiang high-speed railway test, the roll angle and lateral displacement of the center of gravity of the carbody was measured and the roll angle of the car-body relative to the bogie frame was also measured by Eq. (8), as shown in Fig. 20. It can be seen that the car-body roll angle and the roll angle of the car-body relative to the bogie frame test result curves are very similar. This demonstrates a good follow-up characteristic, and there were similar amplitudes when the environmental wind speed was low. This demonstrates that the roll angle of the car-body relative to the bogie frame was much larger than the roll angle of the bogie frame relative to the wheelset under strong wind conditions. Therefore, the transmission path of the wind load acting on the car-body was transmitted from the car-body to the wheel-rail, resulting in a load increase on one side of the train and load shedding on another side. This then caused a follow-up change in the wheel-rail force, and the influence of the environmental wind on train overturning was much greater than other factors, such as track irregularity. In the South Xinjiang railway test, the roll angle, lateral displacement of the center of gravity of the car-body, and the overturning coefficient were measured, as shown in Fig. 21. It can be seen that when the high-speed train passed through the complex terrain under strong wind conditions, the roll angle and lateral displacement of the gravity center of the carbody were consistent with the changing trend of the train overturning coefficient. This indicates that there is a strong linear relationship between the roll angle, lateral displacement of the gravity center of the car-body, and train overturning coefficient. Similarly, this also shows that under strong wind conditions, the transmission path of the wind load acting on the vehicle system was transmitted from the car-body to the wheel-rail. Also, the large-amplitude car-body vibration caused by the environmental wind was the primary reason for the follow-up change in the train overturning coefficient, and the influence of the environmental wind on the overturning safety of the train was much greater than other factors.

Table 2 Overturning coefficients under different cases. Case

D1 D2 D3

Parameters Car-body roll angle (hC ) p p p

Car-body lateral displacement(yC ) p p p

Bogie frame lateral acceleration(ayB ) p p

Wheelset lateral acceleration(ayWs ) p

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D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

1.2

1.2 0.9

P1

D

D1

D2

D3

0.8

P2

D3

D

D

D1

D2 P3

0.4

0.6

D

0.3

0.0

0.0

-0.4

-0.3

-0.8

Wind speed direction reverses 8

9

10

11

12

13

14

15

16

17

18

8

9

10

11

12

13

14

15

16

17

18

Time (s)

Time (s)

Fig. 18. Comparison of the overturning coefficient calculated from the wheel-rail force and the vibration parameters. (a) Case A; and (b) Case B.

P1 P2 P3

CerrD1

CerrD2

CerrD3

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

CerrDi Fig. 19. The CerrD under different conditions.

1.25

1.25

—— Car-body relative to the bogie frame —— Car-body

1.00

0.75

0.75

0.50

0.50

Roll angle (°)

Roll angle (°)

1.00

0.25 0.00 -0.25

0.25 0.00 -0.25

-0.50

-0.50

-0.75

-0.75

-1.00

0

300

600

Time (s)

900

1200

-1.00

—— Car-body relative to the bogie frame —— Car-body 0

10

20

30

40

50

60

70

Time (s)

Fig. 20. Test result of the car-body roll angle and the roll angle of the car-body relative to the bogie frame. (a) Under a higher wind speed condition; and (b) under a lower wind speed condition.

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

Lateral displacement (mm)

16

Roll angle (°)

4.5 3.0 1.5 0.0

70 0 -70

0.9

0.9

0.6

0.6

0.3

0.3

D

D

-1.5

140

0.0

0.0

-0.3

-0.3 0

5

10

15

20

25

30

0

5

10

Time (min)

15

20

25

30

Time (min)

Fig. 21. Car-body vibration displacements and overturning coefficient of the test train: (a) roll angle and overturning coefficient; and (b) lateral displacement of the center of gravity of the car-body and the overturning coefficient.

1.0 0.8

1.0

—— From instrumented wheelset —— From car-body vibration paramters (D3)

0.8 0.6

0.4

0.4

D

D

0.6

—— From instrumented wheelset —— From car-body vibration paramters (D3)

0.2

0.2

0.0

0.0

-0.2

-0.2

-0.4

-0.4

0

5

10

15

Time (min)

20

25

30

11.0

11.5

12.0

12.5

13.0

13.5

14.0

Time (min)

Fig. 22. Comparison of the train overturning coefficient converted from the instrumented wheelset to that obtained from the car-body roll angle and lateral displacement of the center of gravity.

To further verify the feasibility of Eq. (32), the overturning coefficient obtained using Eq. (32) was compared with the overturning coefficient measured using the instrumented wheelset, as shown in Fig. 22. However, due to commercial secrecy, we cannot provide the test train name and basic parameters of this test. In this section, C ¼ 0:1305(the roll angle unit is degrees), E ¼ 1:826(the lateral displacement of the center of gravity of the car-body unit is m), and j ¼ 0:0088. It can be seen that the trends in the two curves are consistent, and the value amplitudes are near to each other. This indicates that it is feasible to predict train overturning safety using the car-body roll angle and lateral displacement of the center of gravity. 5. Conclusions and future work To study the relationship between the car-body vibration and train overturn under strong wind conditions, a method for estimating the train’s overturning coefficient using car-body vibration parameters was studied. The validity of the estimation method was verified by full-scale test. The conclusions are as follows: Under strong wind conditions, there was a strong linear relationship between the roll angle, lateral displacement of the center of gravity of the car-body, and the train overturning coefficient. The train overturning coefficient vibration primary frequency was close to the train’s natural frequency of the car-body lower-center rolling motion. In addition, the vibrations

D. Liu et al. / Mechanical Systems and Signal Processing 142 (2020) 106743

17

that may cause the train to overturn were primarily the car-body large-amplitude and low-frequency lower-center rolling and traverse motions. Under strong wind conditions, the train overturning coefficient was more likely to exceed the critical value than the derailment coefficient. The effect of the bogie frame and the wheelset lateral acceleration on the train derailment was much greater than that on the train overturning. The influence of environmental wind on train overturn was much greater than other factors, such as track irregularity. Under strong wind conditions, the overturn risk during train operation can be predicted using Eq. (25). However, when the roll angle and the lateral displacement are too large, causing a large offset in the pantograph that influences the quality of pantograph-catenary current collection, the maximum values of the car-body roll angle and the lateral displacement in the center of gravity allowed during actual train operation will be smaller than that of the critical train overturning. Car-body vibration under strong wind conditions not only threatens train operational safety but also affects train ride comfort. Therefore, from the perspective of actual train operation, it is necessary to improve the train anti-wind-induced vibration capability by eliminating or reducing vibration in the car-body under strong wind conditions. Therefore, carbody vibration control measures and devices that can be used under strong wind conditions will be studied in the future. Funding The work presented here was supported by the National Key R&D Program of China (2016YFB1200400), the National Natural Science Foundation of China (U1534210) and the China Scholarship Council. CRediT authorship contribution statement Dongrun Liu: Conceptualization, Methodology, Software, Data curation, Writing - original draft. Gisella Marita Tomasini: Writing - review & editing. Daniele Rocchi: Writing - review & editing. Federico Cheli: Writing - review & editing. Zhaijun Lu: Writing - review & editing. Mu Zhong: Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Jiqiang Niu, Yueming Wang, Dan Zhou, Effect of the outer windshield schemes on aerodynamic characteristics around the car-connecting parts and train aerodynamic performance, Mech. Syst. Sig. Process. 130 (2019) 1–16. [2] L. Xu, W. Zhai, Z. Chen, On use of characteristic wavelengths of track irregularities to predict track portions with deteriorated wheel/rail forces, Mech. Syst. Sig. Process. 104 (2018) 264–278. [3] Dongrun Liu, Lu. Zhaijun, Mu. Zhong, et al, Measurements of car-body lateral vibration induced by high-speed trains negotiating complex terrain sections under strong wind conditions, Veh. Syst. Dyn. 56 (2) (2018) 173–189. [4] Dongrun Liu, Mu. Qianxuan Wang, Zhong,, et al, Effect of wind speed variation on the dynamics of a high-speed train, Veh. Syst. Dyn. 57 (2) (2019) 247–268. [5] Dongrun Liu, Lu. Zhaijun, Tianpei Cao, et al, A real-time posture monitoring method for rail vehicle bodies based on machine vision, Veh. Syst. Dyn. 55 (6) (2017) 853–874. [6] Katsuhiro Kikuchi, Minoru Suzuki, Study of aerodynamic coefficients used to estimate critical wind speed for vehicle overturning, J. Wind Eng. Ind. Aerodyn. 147 (2015) 1–17. [7] S. Giappino, D. Rocchi, P. Schito, et al, Crosswind and rollover risk on lightweight railway vehicles, J. Wind Eng. Ind. Aerodyn. 153 (2016) 106–112. [8] CEN. EN14067-6, Railway Applications-Aerodynamics Part-6: Requirements and test procedures for crosswind assessment. European Standard; 2003 [9] Y. Hibino, T. Shimomura, K. Tanifuji, Full-scale experiment on the behavior of a railway vehicle being subjected to lateral force, J. Mech. Syst. Transport. Logist. 3 (1) (2010) 35–43. [10] S. Iwnicki, Handbook of Railway Vehicle Dynamics, CRC Press, 2006. [11] Jiqiang Niu, Dan Zhou, Yueming Wang, Numerical comparison of aerodynamic performance of stationary and moving trains with or without windbreak wall under crosswind, J. Wind Eng. Ind. Aerodyn. 182 (2018) 1–15. [12] Jiqiang Niu, Dan Zhou, Xifeng Liang, Numerical investigation of the aerodynamic characteristics of high-speed trains of different lengths under crosswind with or without windbreaks, Eng. Appl. Comput. Fluid Mech. 12 (1) (2018) 195–215. [13] Dongrun Liu, Wei Zhou, Lei Zhang, et al, Momentary discomfort of high-speed trains passing through complex terrain sections under strong wind conditions, Veh. Syst. Dyn. (2019) 1–24. [14] D. Thomas, M. Berg, S. Stichel, Measurements and simulations of rail vehicle dynamics with respect to overturning risk, Veh. Syst. Dyn. 48 (1) (2010) 97–112. [15] Y. Hibino, H. Kanemoto, Fundamental Study of influence of lateral car-body vibrations on overturning in case of crosswinds, Quart. Rep. RTRI 59 (4) (2018) 249–254. [16] J. Xiang, D. He, Q. Zeng, Effect of cross-wind on spatial vibration responses of train and track system, J. Central S. Univ. Technol. 16 (3) (2009) 520–524. [17] Dongrun Liu, Zhaijun Lu, Tianpei Cao, et al. Experimental study on vibration displacement of a CRH2 EMU under strong wind conditions. 2016 IEEE International Conference on Intelligent Rail Transportation (ICIRT). IEEE, 2016: 377-381. [18] D. Thomas, B. Diedrichs, M. Berg, et al, Dynamics of a high-speed rail vehicle negotiating curves at unsteady crosswind, Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 1 (6) (2010) 1–13. [19] D. Thomas, M. Berg, S. Stichel, Rail,, et al, vehicle response to lateral car-body excitations imitating crosswind, Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 229 (1) (2015) 34–47. [20] GB5599-85: 1985. Railway vehicles-specification for evaluation the dynamic performance and accreditation test, Chinese Railways, Standard leaflet. [21] Futian Wang, Vehicle System Dynamics, China Railway Publishing House, 1994.

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