Effect of Inter-Vehicle Suspension on Variable Speed Curve Running of Train under Crosswinds

Abstract

High-speed trains operating in windy areas may accelerate and decelerate frequently to maintain safe travel, especially when passing curves. During acceleration and deceleration, the role of inter-vehicle suspension (IVS) cannot be ignored. The present study aims to evaluate the effect of IVS on the variable speed curve running of trains under crosswinds. To achieve this purpose, a multibody model of a China Railways High-speed 2 (CRH2) high-speed train considering the IVS is established. By inputting the crosswind loads and traction or braking forces to the model and setting curved tracks with different radii and the unloading factor set as safety criterion, the safe running speeds of the train under different crosswind speeds and different track radii were obtained. The difference in the vehicle dynamics considering the IVS and the fixed connection under traction and braking conditions is analyzed. The radius of the curve track significantly affects the safety characteristics of a train under crosswinds, but its impact diminishes for radii greater than 7000 m. The lateral acceleration, movement angle, unloading factor, and derailment coefficient in both acceleration and deceleration cases of car bodies are affected by the IVS. As a consequence, the IVS will lead to lower safe speeds than fixed connections, but it will also convey more realistic and credible train dynamics.

1. Introduction

In recent years, high-speed rail has developed rapidly around the world. The development of high-speed rail in northwest China is also underway, and the operation of high-speed rail is bound to bring great convenience to cities along the route. However, the operating environment of high-speed trains in northwest China is not optimistic. Wang et al. [1] found that the environment in northwest China is harsh, mostly desert and strong winds. Particularly, Liu et al. [2] found that the Baili wind area, one of the five windy areas through which the Lanzhou–Xinjiang high-speed railway (maximum operating speed of 250 km/h) passes, has large wind speed changes, which brings great challenges to the safe operation of high-speed trains. Liu et al. [3] found that the change in wind speed along the railway line is particularly serious in windy areas, and according to incomplete statistics, more than 170 sections of the Lanzhou–Xinjiang high-speed railway line exhibit the phenomenon of “car swaying”. According to Yao et al. [4], strong crosswinds caused at least 26 serious train safety accidents from 1956 to 2000 on the Lanzhou–Xinjiang railway. Zhou et al. [5] found that to maintain safety under crosswinds, trains need to slow down to pass safely and then accelerate to the normal speed, which may result in frequent acceleration and deceleration processes. As shown in Figure 1, when the train passes through curved tracks under crosswinds, the possibility of derailment and overturning is greatly increased, and the safety risk is greatly increased.

A large number of scholars have performed studies on the safety of trains under crosswinds. Liu et al. [6] focused on the effect of changing wind speeds on a train’s operational safety and obtained many important research results. Liu et al. [3] analyzed the roll angle and lateral displacement of car-body gravity by performing a field measurement along the Lanzhou–Xinjiang high-speed railway in China. Montenegro et al. [7] evaluated the derailment risk of an HS train crossing the future Volga River railway bridge by using a stochastic wind model and found that the unloading ratio was most critical for train speeds below 300 km/h. Zhang et al. [8] analyzed the dynamic of metro vehicles traveling on a high-pier viaduct under cross-wind in Chongqing. Thomas et al. [9] verified the accuracy of the train dynamic model by a real vehicle test. They studied the dynamic response of the train passing through the curve track under changing wind speed cases by combining it with the gust model. You et al. [10] studied the influence of Chinese hat gust wind on the passing dynamics of train curves through wind tunnel tests and dynamic simulations. And they found that the wind direction reversal had a great impact on the safety of train overturning.

Vehicle system dynamics has developed rapidly in recent decades, and many scholars have published a large number of related research papers and established mature vehicle-dynamics models, including single-vehicle models and multi-vehicle models [11,12,13,14]. A multi-section vehicle-dynamics model is used in this article.

Most of the world’s high-speed trains are designed with inter-vehicle shock absorbers, tight-lock devices, and inter-vehicle windshields, such as Japan’s Shinkansen, Germany’s Inter City Express 3, and China’s CRH2. The purpose of these devices is to absorb vibrations between vehicles and improve the overall dynamics of the vehicle. Therefore, when the train is running in a complex environment, such as a crosswind area, the role of the IVS device cannot be ignored when studying the vehicle-dynamics problems of multi-car trains.

The train must slow down to reach safe travel criteria when passing through a drastically changing wind zone and then accelerate back to normal travel speed when the wind speed decreases. Most of the previous scholars studied the train dynamic response just considering the single-vehicle dynamic model and did not consider the longitudinal dynamics between vehicles. Or, they only considered the fixed connections (without IVS) between vehicles to study entire-train dynamics, which may ignore the influence of IVS on train movement, especially the vibration acceleration of the car body. Ling et al. [15] established a 3D dynamic model of a high-speed train coupled with a flexible ballast track and considering nonlinear couplers and inter-vehicle dampers, and the linear tight-lock vestibule diaphragm was established to simulate the effect of the end connections of neighboring vehicles on dynamic behavior; it was found that inter-vehicle connections have an important influence on the dynamic behavior of high-speed vehicles. Ling [16] studied the longitudinal dynamic behavior of high-speed trains under variable speed by establishing an eight-vehicle train model considering the IVS. However, he did not take into account the curve-passing condition and the influence of crosswinds. Therefore, it is necessary to study the dynamic response and safe driving standards of high-speed trains considering the real IVS characteristics in complex operating environments. It would be advantageous to incorporate the actual IVS characteristics into the vehicle-dynamics model as a nonlinear force element and to evaluate the safety properties of the train under various radii and operating conditions such as traction and braking.

Therefore, based on the above engineering background requirements and related research, we established a three-vehicle-dynamics model considering the IVS by using the multi-rigid body dynamics method. The dynamic response of the vehicle when accelerating and decelerating through a curve track under crosswinds with different speeds was studied. The vibration acceleration of the three carriages with or without IVS during traction and braking was analyzed. The physical model of the vehicle, the setting of boundary conditions such as the multibody dynamics model and traction braking force, as well as the validation of the multibody dynamics model of the vehicle, are all in Section 2. Whether to consider IVS for the simulation of high-speed train dynamics under different wind speeds, vehicle speeds, and track radius conditions is discussed in Section 3. The conclusions are drawn in Section 4. In this study, we studied the safe speed and dynamic response of the CRH2 train considering IVS devices in a complex environment. At the same time, the vibration responses of different carriages during the frequent speed changes of the train, as well as the influence of the IVS, are obtained. This provides an important reference for studying the safety of high-speed trains in complex operating environments and provides a more realistic safe operating speed. It can also help researchers to better understand the role of IVS devices in the process of train speed change and provide truer references for the safety basis of the curve passing of the train under crosswinds.

2. Establishment and Definition of Train Model

2.1. Train Dynamics Method

Based on the establishment of the three-vehicle multibody dynamics model, we study the safety indicators such as unloading factor, derailment coefficient, and the ride quality such as car-body roll angle and lateral acceleration of the train body. The basic process is shown in Figure 2 and involves inputting aerodynamic loads of crosswinds and traction or braking force into the multibody dynamics model of the train with two different cases of connections, then calculating and outputting the dynamic results.

2.2. Schematic Diagram of the Curve Passing of the Vehicle

The train model established in this paper is based on a real-scale CRH2 high-speed train running on the Lanzhou–Xinjiang high-speed railway in China. Figure 3a shows a schematic diagram of the train running under crosswinds. Figure 3b shows the train line, which is a curve track consisting of a straight track, spiral transition curve track, circle track, spiral transition curve track, and straight track.

2.3. Wind Aerodynamic Forces

The wind is classified into two types: mean and fluctuating winds. Mean winds are equivalent to static forces, while fluctuating winds represent turbulence and randomness in the atmospheric boundary layer. Turbulence is a complex problem in fluid dynamics that can regulate the vertical exchange of momentum, energy, and composition of the wind. The structure of the wind field caused by the train is complex, and the turbulence problem is inevitable. There are many studies on the turbulent atmospheric boundary layer [17,18,19,20,21,22]. Large-scale detailed calculations of the flow field around trains require many computational resources. Therefore, some researchers use the quasi-steady calculation method to calculate the aerodynamic load around high-speed trains [23]. Li [24] found that the aerodynamic characteristics around the train calculated by the Chinese hat wind model were in good agreement with the wind speed. Liu et al. [6] discussed the error of the two calculation methods and found that the error between the quasi-static calculation method and the unsteady calculation method was small. Therefore, a quasi-static method was used to calculate the wind load. The effects of turbulence are not considered.

In this paper, to simplify the calculation, a steady wind in the time domain was used to describe the crosswinds, and wind loads are simulated as concentrated forces and moments. As is shown in Figure 4, the wind loads were exerted on the pneumatic center of the head car, middle car, and tail car. The specific calculation methods are given in Equations (1) and (2).

$$F_i=\frac{1}{2}\rho S_c{V}_{res}^2{C}_{Fi}\left(\beta \right)$$

(1)

$$M_i=\frac{1}{2}\rho S_c{L}_c{V}_{res}^2{C}_{Mi}\left(\beta \right)$$

(2)

$$V_{res}^2=V_c^2+V_w^2-2V_c{V}_{w}cos\left(\pi -\alpha \right)$$

(3)

where ρ is the air density, S_c is the area of the vehicle affected by crosswinds; C_Fi (β), and C_Mi (β) are aerodynamic coefficients that depend on the yaw angle β, which we can get from Liu et al. [2]. L_c is the height of the car. V_res is the relative speed of the wind to the train.

The relationship between synthetic composed wind speed and vehicle speed is shown in Figure 5 and calculated in Equation (3) where V_c is the speed of the train, V_w is the speed of the wind, and α is the wind attack angle.

2.4. Dynamic Model of Train

A real and correct train dynamics model is the premise of studying the train dynamic responses. We also simplified the model to reduce the computational resources and duration. According to the vehicle-dynamics method, a single vehicle consists of a car body, bogies, wheelsets, primary suspension, second suspension, and axle box, ignoring their elastic deformation. The track model was considered as an inertia fixed rail.

The multibody dynamics of the CRH2 high-speed train are shown in Figure 6 [15]. Specifically, Figure 6a displays the train’s side view, and Figure 6b provides its top view. A single vehicle contains a car body, two bogies, four wheelsets, and eight axle boxes. Each rigid body has six degrees of freedom (DOFs), but the axle box has only the pitch DOF. The degrees of freedom of the IVS device are not taken into account, but they are treated as force elements with nonlinear mechanical characteristics; the two ends of the IVS device model are fixed in the center of the base of the end of the adjacent two car bodies. Hence, a single-vehicle train has a total of 50 DOFs and a three-vehicle train has a total of 150 DOFs. The primary suspension that connects the wheelset with the bogie contains a helical spring, a vertical damper, and an axle box. The second suspension that connects the bogie with the car body contains two air springs, two lateral dampers, two anti-yaw motion dampers, a lateral bump stop, an anti-roll bar, and a traction rod. A coupler buffer device and two longitudinal dampers were considered to be the IVS. The mechanical property is simulated by a nonlinear spring-damper element. These suspensions were installed symmetrically, and their mass was ignored. The DOFs of the vehicle component are shown in

Table 1. DOFs of the vehicle component.

Vehicle Component	DOFs of the Vehicle-Dynamics Model
	Longitudinal	Lateral	Vertical	Roll	Pitch	Yaw
Car body	Xc	Yc	Zc	Φc	βc	Ψc
Bogie F	XbF	YbF	ZbF	ФbF	βbF	ΨbF
Bogie B	XbB	YbB	ZbB	ФbB	βbB	ΨbB
Wheelset FF	XwFF	YwFF	ZwFF	ФwFF	βwFF	ΨwFF
Wheelset FB	XwFB	YwFB	ZwFB	ФwFB	βwFB	ΨwFB
Wheelset BF	XwBF	YwBF	ZwBF	ФwBF	βwBF	ΨwBF
Wheelset BB	XwBB	YwBB	ZwBB	ФwBB	βwBB	ΨwBB
Axle box (i = 1–8)					βboxi

. The primary parameters adopted for the CRH2 dynamic model are available in Liu et al. [2].

2.5. The Motion Differential Equations of the Vehicle Dynamic System

The displacement vector of the vehicle can be described in Equation (4):

$${\mathit{X}}_v^T=\left\{{\mathit{X}}_{c,\left(1-3\right)}^T,{\mathit{X}}_{b,\left(1-6\right)}^T,{\mathit{X}}_{a,\left(1-24\right)}^T,{\mathit{X}}_{w,\left(1-12\right)}^T\right\}$$

(4)

where X_c, X_b, X_a, and X_w are the displacement vectors of the car bodies, bogies, axle boxes, and wheelsets, respectively. The subscript number represents the designator of the corresponding component, and T is the transpose operator. By the Lagrange method, the motion differential equations of the vehicle dynamic system be expressed as Equation (5):

$$M_v{\ddot{\mathit{X}}}_v+C_v{\dot{\mathit{X}}}_v+K_v{\mathit{X}}_v=F{a}_v$$

(5)

where M_v is the inertia matrix, and C_v is the damping matrix, which is determined by the damping of the springs and dampers. K_v is the stiffness matrix, which depends on the spring stiffness. Fa_v is the resultant force of the aerodynamic force on the train caused by crosswinds.

2.6. Wheel–Rail Model and Contact Setting

In this paper, the wheelset radius is 0.43 m, and the wheel and rail profile types are LMA and CHN60, respectively. The normal contact forces between the wheel and rail are computed through the nonlinear Hertz contact theory (Hertz [25]), and the tangential contact forces are computed through Kalker’s theory (Kalker [26]). Since the main source of excitation of the vehicle under crosswinds is the wind loads, and the track irregularity is not considered in European standard EN 14067-6 [27], the track irregularities are not considered in this paper.

2.7. Characteristics of the IVS

In this paper, the train model is considered with the IVS, so the train model is more in line with the actual dynamic characteristics of the train.

According to the impact test of the coupler buffer device of the CRH2 vehicle conducted by Zhao et al. [28], the data obtained from the test were input into the force element used to simulate the coupler buffer device and consider the hysteresis effect and backlash. The nonlinear characteristics curves of the coupler buffer device are shown in Figure 7a, and the characteristics of the dampers between vehicles are shown in Figure 7b. In this paper, the IVS is considered as a hook buffer and two dampers between the carriages, as shown in Figure 7c. The photograph of the dampers and buffers is shown in Figure 7d [29].

2.8. Traction and Braking Forces

Ling [16] found that a three-vehicle-dynamics model can already reflect the entire dynamic characteristics of the high-speed train. Therefore, a three-vehicle-dynamics model was established in which the head car and tail car were set as motor cars, and the middle car was a trailer. The traction and braking power of the train are complex in actual operation, and this paper only uses the equivalent constant force to simulate the traction and braking forces. The equivalent constant traction and braking force are shown in Figure 8 and available in Ling [16]. For ideal train acceleration conditions, all 8 wheelsets of the head and tail cars are given the same traction force of 5750 N. Therefore, the total traction of the vehicle is F_Acc = 46,000 N; for ideal train deceleration conditions, the total braking force F_Dec= −89,371 N is evenly distributed across 12 wheelsets of the train. According to Newton’s second law, the theoretical acceleration and deceleration of trains are given in Equations (6) and (7). However, in the simulation model, the actual acceleration of the vehicle will be smaller than the theoretical value due to the friction between wheels and rails, components, dampers, and air friction.

$$a_1=\frac{F_{T1}}{M_T}=\frac{8F_{Acc}}{M_1+M_2+M_3}=0.335926 \mathrm{m}/{\mathrm{s}}^2$$

(6)

$$a_2=\frac{F_2}{M_T}=\frac{12F_{Dec}}{M_1+M_2+M_3}=-0.65265\mathrm{m}/{\mathrm{s}}^2$$

(7)

2.9. Model Verification

To verify the dynamic model, input the same gust model shown in Figure 9 as Liu et al. [2]. Where V_max = 30 m/s, V_mean = 15 m/s, the maximum wind velocity duration Δt = 6 s.

It can be seen that the unloading factor in Figure 9a and the maximum roll angle in Figure 9b both have good consistency with Liu et al. [2]. However, due to the inconsistency of the set force elements, the vibration attenuation of this model may be faster. Liu et al. [2] validated the numerical simulations with wind tunnel tests, and therefore, the train model in this study is proven to be credible by previous research.

2.10. Safety Assessment

In this paper, we set the unloading factor and derailment coefficient as safety criteria. The lateral acceleration of the car body and the movement angle of the car body are used for stability evaluation. Referring to EN 14067-6 [27], the unloading factor (D) is determined by the expression of the vertical wheel–rail force Q and the static vertical wheel–rail force Q₀. The derailment coefficient (T) is the ratio of the tangential force Q_d to the vertical wheel–rail force P_d. Both D and T can reflect if the wheel is unloading and are calculated by Equations (8) and (9).

$$D=\frac{Q_0-Q}{Q_0}<0.8$$

(8)

$$T=\frac{Q_d}{P_d}<0.8$$

(9)

3. Multibody Simulations

3.1. Setting of Curved Lines and Vehicle Speeds

Reasonable different curve radii and super-elevation should be set to study the curve-passing ability of CRH2 under different curve radii. The curve radius is set from R4500 m to R9000 m, and details can be found in

Table 2. Parameters for curve lines.

Track Radius(m)	V (km/h)	4500	5000	6000	7000	8000	9000
Spiral transition curve length(m)	100–250	340	300	250	220	190	170
Circle length(m)		200	200	200	200	200	200
Super-elevation(mm)		105	105	105	105	105	105
Spiral transition curve length(m)	300	510	440	410	350	300	270
Circle length(m)		200	200	200	200	200	200
Super-elevation(mm)		151	151	151	151	151	151
Spiral transition curve length(m)	350	590	590	590	590	530	470
Circle length(m)		200	200	200	200	200	200
Super-elevation(mm)		175	175	175	175	175	175

. Since this paper does not focus on the influence of super-elevation, the balance super-elevation calculated at R7000 m and V250/300/350 km/h is selected as the balance super-elevation at different radii, and the calculation method is shown in Equation (10).

$$h=\frac{v^{2}S}{{3.6}^{2}Rg}$$

(10)

where v is the speed of the vehicle passing through the curve track, S is the center distance of the standard rail (1500 mm), R is the radius of the curve, and g is the acceleration of gravity (9.81 m/s²). The maximum operating speed of the Lanzhou–Xinjiang high-speed railway is 250 km/h, and the maximum travel speed of CRH2 trains is 350 km/h. To study the safety of the train curve passing more comprehensively, the train speed range is set to 100–350 km/h.

3.2. Security of Train Curve Passing at a Constant Speed

Figure 10 shows the influence of curve radius R and wind speed (V_w) at different vehicle speeds (V_c) on the unloading factor. When V_c is less than 250 km/h, the unloading factor changes a little when the radius increases, and after V_c is greater than 250 km/h, the unloading factor slowly decreases as the radius increases. When the V_w is 15 m/s or less, the vehicle is safe in the range of 0–350 km/h. When the V_w reaches 30 m/s and the V_c reaches 200 km/h, the unloading factor has exceeded the safety limit value of 0.8, and the possibility of the train capsizing is greatly increased; when the V_w is 25 m/s, and when the V_c exceeds 250 km/h, there is a risk of overturning when the train curve passes; when the V_w is 20 m/s, when the V_c reaches 350 km/h, and the radius is less than 5000 m, the unloading factor has exceeded the safety limit value 0.8. When the V_c is greater than 250 km/h and the V_w is greater than 30 m/s, the train wheel sets have been derailed and should stop driving.

As shown in Figure 11 and Figure 12, the relationship between the safety vehicle speed and wind speed under the same radius can be obtained. The maximum vehicle speed does not exceed 350 km/h. As can be seen in

Table 3. Safe speed in two cases.

		Wind 15 m/s	Wind 20 m/s	Wind 25 m/s	Wind 30 m/s
With suspension	R4500 m	safe	330 km/h	234 km/h	169 km/h
	R7000	safe	safe	245 km/h	171 km/h
	R9000	safe	safe	246 km/h	175 km/h
Fixed connection	R4500	safe	safe	252 km/h	168 km/h
	R7000	safe	safe	254 km/h	169 km/h
	R9000	safe	safe	260 km/h	173 km/h

, as the radius increases, the safe speed of vehicle curve passing increases, but the difference between R7000 and R9000 is not much. Additionally, the maximum safe speed is reduced when considering the IVS. For example, When the wind speed is 25 m/s and the radius is 9000 m, 7000 m, and 4500 m, considering the IVS will reduce the maximum safety speed by 14 km/h, 8 km/h, and 18 km/h, respectively. When the wind speed is 20 m/s and the radius is 4500 m, the case of fixed connection is safe, but considering the IVS will limit the safe speed to 330 km/h. Moreover, considering the IVS has little influence on the safety speed of the vehicle when the wind speed is 30 m/s, probably due to the safety speed being small in both cases.

3.3. Dynamic Responses of Vehicle When Decelerating through Curved Tracks

When the train is traveling in strong wind conditions, for example at a wind speed of 25 m/s, according to the above analysis, it can be seen that the maximum critical safety speed of the train is 246 km/h. Therefore, it is necessary to decelerate for safety. To study the influence of the IVS on the dynamic characteristics of the train decelerating curve passing, the initial working conditions of initial velocity V₀ = 250 km/h, R7000 m, and V_w = 25 m/s were set.

Figure 13 shows the six bogies from the head car to the tail car, the front bogie (FF), the rear bogie (FB) of the head car, the front bogie (MF), the rear bogie (MB) of the middle car, the front bogie (BF) and the rear bogie (BB) of the tail car. The error calculation method in Figure 13 is shown in Equation (11). P_Non is the unloading factor in the case of a fixed connection, and P_With is the unloading factor in the case of considering the IVS.

$$\mathrm{Error}=\left(P_{Non}-P_{With}\right)/P_{Non}$$

(11)

It can be seen that the FF bogie has the highest unloading factor and derailment coefficient of all bogies during train deceleration. Considering the IVS, the unloading factor of the MB and BF bogies is reduced by 2.89% and 3.38%, respectively. The derailment coefficients of the MB, BF, and BB bogies were reduced by 2.25%, 5.66%, and 2.13%, respectively. This is probably because the braking force is uniformly distributed over all wheelsets, and the IVS absorbs the impact loads between the vehicles, thus reducing the unloading factors and derailment coefficients and improving travel safety. In addition, the unloading factors of the vehicle always reach a dangerous value before the derailment coefficient, which is consistent with the study of Zhai et al. [30].

Figure 14 shows the lateral acceleration of the head, middle, and tail car bodies during deceleration, and it can be seen that there are four higher peaks in the time domain, and these four peaks correspond to the four transition areas of the curved track section. The lateral acceleration of the head car and tail car are only slightly affected by the IVS. However, considering the IVS reduces the maximum lateral acceleration of the middle car by 49.02%, it shows that the IVS effectively suppresses the lateral vibration of the middle car.

As shown in Figure 15, the body roll angle is the largest of the body motion angles during train deceleration in crosswind conditions. The IVS had little effect on the maximum roll angle of the head car F and middle car M; however, it reduced the maximum roll angle of the tail car by 5.08%, considering the IVS increases the maximum pitch angle of the middle car M by 8.11%. This is because compared to the fixed connection, the coupler buffer device between the vehicles can absorb the energy of the impact and the existence of backlash between the vehicles allows a certain displacement, so the pitch angle of the middle car increases during deceleration. Similarly, the IVS has little effect on head car F and tail car B but reduces the maximum yaw angle of the middle car by 14%. This is because the dampers between the vehicles work to dampen the yaw motion between the vehicles effectively.

3.4. Dynamic Responses of Vehicle When Accelerating through Curved Tracks

When the wind speed decreases (below 15 m/s) and the vehicle speed is below 150 km/h, the train should be re-accelerated to return to operating speed. Therefore, to study the influence of the IVS on the dynamic characteristics of the train re-accelerating through curved tracks, the initial working conditions of initial velocity V₀ = 150 km/h, R7000 m, and V_w = 15 m/s were set.

As can be seen from Figure 16a, the unloading factors of the four bogies FF, FB, MF, and BF are increased when the IVS are taken into account under acceleration conditions. Figure 16b shows that considering the IVS increases the maximum derailment coefficient for all bogies except the FF bogie. Of the six bogies, the FF bogie has the highest unloading factor of all bogies. Considering the IVS, the maximum values of the unloading factor for FF, FB, MF, and BF bogies were increased by 0.11%, 1.05%, 0.36%, and 7.27%, respectively. Similarly, the FF bogie has the maximum value of derailment coefficient. Considering the IVS, the maximum values of the derailment coefficient for FB, MF, MB, BF, and BB bogies were increased. Particularly, the BF bogie was increased by 6.94%. Since only the head car and tail car are motor cars during acceleration, the longitudinal acceleration of the car bodies will be different. The motion and impact effects between the vehicles will be more realistic if the suspension between the vehicles is considered rather than a simple fixed connection. Therefore, considering IVS will slightly increase the unloading factor and derailment coefficient. The safe speed of the vehicle will be reduced, and the results will be closer to the real situation.

It can be seen from Figure 17a that considering the IVS, the peak of the lateral vibration acceleration of the head car when driving out of the spiral transition curve track is reduced by 55.13%. Considering the IVS, the peak of the lateral vibration acceleration of the middle car is reduced at all four transitions in the curve track. However, as shown in Figure 17c, the peak of the lateral vibration acceleration of the tail car when entering the spiral transition curve track is decreased by 68.17%. For the same reason, only the head car and tail car are motor cars during acceleration, and the longitudinal accelerations of the head car and tail car are different from the middle and cause more impact between the vehicles. In the case of fixed connections, the motion between the bodies is ignored. Hence, the lateral acceleration of the vehicle bodies will be larger but more realistic when considering the IVS.

As shown in Figure 18, the body roll angle is the largest of the body motion angles during train acceleration in crosswind conditions. The IVS has little effect on the maximum roll angle of the head car F and middle car M. However, it increased the maximum roll angle of the tail car by 6.05%. Considering the IVS has a great influence on the maximum value of pitch angle of the three car bodies. The pitch angle of the head car, the middle car, and the tail car increased by 39.71%, 8.6%, and 130.89%, respectively. This is due to the existence of backlash in the coupler buffer device, but the fixed connection does not allow for relative displacement between the car bodies. For the yaw angle of the car body, considering the IVS will reduce the yaw angle of car F and car M by 4.70% and 37.59%, which is also due to the fact that dampers between the vehicles can suppress the yaw motion between the car bodies.

4. Conclusions

In this study, the effect of the IVS on the dynamic performance of a high-speed train is analyzed by creating force elements to simulate the mechanical properties of the suspension devices between vehicles. A multibody model of a CRH2 high-speed train considering the IVS is constructed. Considering different crosswind wind speeds and different curved lines, the safe driving speed is determined and dynamic differences between the two cases by considering the IVS and fixed connection when the curve passes in acceleration and deceleration conditions under crosswinds are analyzed. The main conclusions can be summarized as follows.

• The safety of the train is sensitive to the radius of the curve track when the curve passes under crosswinds. When the vehicle speed is less than 250 km/h, the unloading factor changes little with increasing radius. When the vehicle speed is 250–350 km/h, the maximum value of the unloading factor reduces with increasing radius. However, the reduction will be little when the radius exceeds 7000 m. Moreover, the safe speed of the train should be less than 250 km/h when the wind speed exceeds 25 m/s.
• Compared with fixed connections, taking the IVS into account will reduce the critical safety speed at the same radius. This reflects more realistic train dynamics. The safe speed decreases by 18 km/h, 9 km/h, and 14 km/h at a wind speed of 25 m/s at the radius of 4500 m, 7000 m, and 9000 m, respectively. The difference in safe speed between the two cases is very small when the wind speed is 30 m/s. Consequently, the true safe speed limit for CRH2 trains should be lower than the current recommended values.
• Considering the IVS will reduce the unloading factor when passing a curve during deceleration under crosswinds. The BF bogie has the greatest reduction, of which the unloading factor is reduced by 3.38% and the derailment coefficient is reduced by 5.56%. Furthermore, the maximum value of lateral acceleration in the middle car is reduced by 49%. For the car-body motion angle, the head car and tail car are insensitive, but the pitch angle of the middle car is increased by 8.11% and the yaw angle is reduced by 14%. Taking into account the IVS, vibrations in the train car are reduced during operation. The level of reduction varies from carriage to carriage.
• Considering the IVS will increase the unloading factor when passing a curve during acceleration under crosswinds. The BF bogie has the greatest increase, of which the unloading factor increases by 7.27%. The derailment coefficient of FB, MB, and BB bogies increases by 4.51%, 3.76% and 6.94%, respectively. Moreover, the maximum value of the lateral acceleration of the head car when driving out of the spiral transition curve track is reduced by 55.13%, and the middle car is also reduced by 32.26% when driving out of the circle track. However, the maximum value of the lateral acceleration of the tail car increases by 68.17% when entering the spiral transition curve track. These results suggest that considering the IVS effectively reduces the maximum value of the lateral acceleration of the vehicle when passing the transition areas of the curved track, especially for the middle car.

In conclusion, considering the IVS will result in lower safe speeds than fixed connections but reflect more realistic and credible train dynamics. However, due to the workload, we did not consider the effect of wind speed variation and only used constant forces to simulate the traction and braking forces. In the future, the effect of wind speed variation and more realistic traction and braking forces will be considered to further deepen the study. Further exploration of the impact of varying IVS mechanical properties on vehicle dynamics will lead to the identification of optimal IVS mechanical parameters.