{'fraction_non_alphanumeric': 0.025404732254047324, 'fraction_numerical': 0.007876712328767124, 'mean_word_length': 4.77923713566031, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nonlinear hunting stability of high-speed railway vehicle on a curved track under steady aerodynamic load

ABSTRACT This paper investigates the nonlinear hunting stability of a high-speed vehicle on a curved track under steady aerodynamic load. We first established a nonlinear dynamic model of high-speed vehicle on a curved track while considering the effect of aerodynamic load. Then, we wrote a numerical simulation programme and verified the validity. The following two types of aerodynamic conditions were concerned in this study: considering only the aerodynamic lift and considering the crosswind loads. The influence of aerodynamic load on the creep force, the restoring force and moment generated by gravity, and the equilibrium position of the vehicle on the curved track can all change the hunting stability. We calculated bifurcation diagrams, limit cycle motions, nonlinear critical speeds, and frequency spectrum considering those two types of aerodynamic conditions, then compared with those without considering the aerodynamic effects. The results show that multiple frequencies including double frequency, triple frequency, quadruple frequency, and high frequencies will appear in addition to the fundamental frequency when hunting motion occurs. The higher frequencies which are even times of fundamental frequency are caused by the deviation of equilibrium position from track centre, while the deviation of equilibrium position is generated by curve track or crosswind load. Additionally, a positive aerodynamic lift and a crosswind load in the centrifugal direction will decrease the vehicle’s motion stability, while a negative aerodynamic lift and a crosswind load in the centripetal direction can improve the stability.


Introduction
A railway vehicle moving along a track is a non-linear system that includes many coupled factors. It is very important to investigate the stability of the non-linear hunting motions of such a vehicle. There have been many important studies of the hunting stability of trains. True [1] provided an accurate definition of the non-linear critical speed and pointed out that this speed can be determined numerically by using the path-following method [2,3].
CONTACT Xiao-Hui Zeng zxh@imech.ac.cn Polach and Kaiser [4] used two methods to analyse the non-linear hunting motion of a railway vehicle. Di Gialleonardo et al. [5] analysed the effects of different types of rail models on bifurcation diagrams. Zeng et al. [6] studied the gyroscopic effect of the wheelsets on the hunting stability of high-speed railway vehicles. Most of these studies focused on hunting stability on a straight track, but railway vehicles on a curved track experience very different operating conditions. Consequently, the non-linear stability of a vehicle on a curved track has also received attention. Using numerical simulations to produce bifurcation diagrams for such a system, Zboinski and Dusza [7,8] investigated the effects on the non-linear stability of the suspension parameters, the wheel/rail profile, wheel/rail wear, the radius of curvature of the track and the rail slope. Their studies [9] explained the cause of a train's self-excited vibrations and elucidated the similarity between a curved track and a straight track. Zeng and Wu [10] analysed the influence of the superelevation of the outer rail and of the radius of the curve on the non-linear critical speed. True et al. [11] calculated the critical speed of the curved track and studied the existence of multiple stable solutions of the curved-track problem. These studies have accordingly provided us with a theory of the non-linear stability of railway vehicles on curved tracks and also with appropriate numerical methods for calculating the non-linear critical speed. Generally, the aerodynamic load is proportional to the square of the wind speed, and the aerodynamic load on a train generated by the reverse airflow and by crosswinds is considerably greater at higher speeds than at lower speeds. For this reason, it is crucial to investigate the aerodynamic loading effect on high-speed trains. Baker et al. [12,13], Yao et al. [14] and Sun et al. [15] conducted aerodynamic and flow-field analyses of high-speed trains. The analysis of the dynamic response and safety factors of high-speed train, aimed at aerodynamic loading conditions, has gradually taken root. Yu et al. [16] and Liu et al. [17] analysed the influence of aerodynamic loads on the dynamic response, derailment coefficient, the wheel load-reduction rate and the overturn coefficient. Cheng et al. [18,19] and Baker et al. [20] investigated the dynamic response and derailment coefficient of a railway vehicle under aerodynamic conditions. Mao [21] discussed the problem of vehicle control under crosswinds.
In summary, most numerical or experimental studies of the aerodynamic effects on high-speed railway-vehicle motions have been performed to obtain the aerodynamic load on the vehicle. The vehicle is then subjected to this load, and the dynamic response is analysed to obtain the derailment coefficient, load-reduction rate and derailment coefficient for the railway vehicle. These are actually forced-vibration problems. In contrast, the hunting stability of a railway vehicle subject to aerodynamic loads is a self-excitation vibration problem, the mechanism of which is different from that of forced vibrations. In other words, the results of existing studies of the aerodynamic loads on a vehicle's forced vibrations are not sufficient to characterise the stability of a vehicle's hunting motions under the influence of an aerodynamic load. It is therefore necessary to study the hunting stability of highspeed railway vehicles that are subject to aerodynamic loading effects. We have previously studied the linear stability of the aerodynamic effects on vehicles travelling on straight and curved tracks [22][23][24][25], and from these investigations we have gained considerable understanding of aerodynamic effects on the hunting stability of high-speed railway vehicles. A steady aerodynamic load can change the gravitational restoring force (and moment) and the creep force (and moment), as well as the equilibrium position of the moving vehicle. In the equations that describe vehicle motions, these effects appear as changes in the stiffness and damping, which implies that the intrinsic characteristics of the vibration system of the high-speed rail vehicle have also changed.
In the present study, we have investigated the non-linear stability of a vehicle on a curved track, extending our previous linear-stability study. To understand the mechanism by which aerodynamic loading influences the hunting stability of a railway vehicle on a curved track, we consider here two aerodynamic conditions: steady aerodynamic lift and crosswind loads. Here, 'crosswind load' refers to the aerodynamic force and moment generated by a combination of high-speed reverse air flow and crosswinds on the train; this includes the lateral aerodynamic force, lift force, roll moment, pitch moment and yaw moment.
We first establish a non-linear dynamic model for a railway vehicle on a curved track that takes into account the aerodynamic load. We have developed a simulation programme based on this model, and we have verified the accuracy of this programme. Using this simulation programme, we have used a path-following method to calculate bifurcation diagrams for a railway vehicle that include the effects of aerodynamic lift and crosswind loads. By analysing the bifurcation characteristics, the non-linear critical speed and limitcycle motions, we have determined the influence of aerodynamic effects on the non-linear hunting stability. In addition, we have analysed the frequency spectrum and determined the high-frequency characteristics of hunting motions on a curved track and under a crosswind load. Figure 1 shows a dynamic model of a vehicle on a curved track, where C 0 is the superelevation of the outer rail, and the numbers represent the elements of the suspension system.

Equations of motion
To investigate the hunting stability of a high-speed railway vehicle that is negotiating a curved track, we have considered 23 rigid-body degrees of freedom of the vehicle: the lateral displacement y w and yaw ψ w of the wheelset (×4), the lateral displacement y t , vertical displacement z t , roll φ t , yaw ψ t and pitch β t of the frame (×2) and the lateral displacement y c , vertical displacement z c , roll φ c , yaw ψ c and pitch β c of the car body (×1). These are collectively expressed in the vector Y 1 : The roll φ w and vertical displacement z w of the wheelset are not independent; they are constrained by the relative lateral displacement and the geometric relationship of the wheel and rail.
In addition, both the secondary lateral damping and the yaw damping are modelled by series-connected springs and dampers, and there are lateral displacements y h of the spring-damped connecting points of the secondary lateral dampers (×2) and longitudinal displacements y s of the spring-damper connecting points of the secondary yaw dampers (×4). These degrees of freedom can be expressed collectively by the vector Y 2 : Y 2 = y h1 y h2 y sL1 y sR1 y sL2 y sR2 T .
The equations of motions for this dynamical model can be written as follows: where M, C and K are the inertia matrix, damping matrix and stiffness matrix, respectively, and P g and P f denote the centrifugal and aerodynamic forces, respectively. The individual governing differential equations are listed Appendix A, and all the physical quantities in Appendix A are defined in Appendix B. The nominal design parameters of the vehicle are also provided in Appendix B.

Wheel/rail geometric profile
The non-linearity of the wheel-rail relationship is one of the most important non-linear factors of the railway-vehicle vibration system. In this paper, we use a combination of the tread profiles of the LMA wheel and the UIC60 rail.

Wheel and rail creep force
According to the Kalker theory of linear creep [26], the wheel/rail creep coefficients are as follows: where R w , r w , r r , G, C ij , m and n are the rolling radius of the wheelset, the transverse radius of the wheel profile, the transverse radius of the rail profile, the shear modulus, the non-dimensional Kalker coefficients and the coefficients about A and B, respectively. The coefficients C 11 , C 22 , C 23 and C 33 are tabulated elsewhere [27]. We calculate the creep force and creep moment according to Kalker's linear creep theory: where f 11 , f 22 , f 23 , and f 33 are creep coefficients, and γ x , γ y , and γ s are creepages.
To calculate the wheel/rail creep force precisely, we used the Shen-Hedrick-Elkins nonlinear creep model [28] and made the following corrections to the creep force: where f is the coefficient of friction between the wheel and the rail. Introducing the correction factor ε defined by we can write the corrected creep force and creep moment as:

Wheel/rail normal force
The expression for the normal force at the wheel-rail contactcan be obtained from the equations for the rolling and vertical motions of the wheelset: Here, N L and N R are, respectively, the normal forces at the left and right contact points; λ L and λ R are, respectively, the contact angles of the left and right wheels; R L and R R are, respectively, the radii of the rolling circles of the left and right wheels; and the quantities F pzL , F pzR , f pzL , f pzR, F yL and F yR are, respectively, the changes in the primary-suspension forces at the left and right sides of the wheelset, the primary-suspension damping forces at the left and right sides of the wheelset and the corresponding lateral creep forces. In Equation (10), we calculate the roll φ w and vertical displacement z w of the wheelset from the geometric relationship between the wheel and rail and the lateral displacement of the wheelset. The normal force at the wheel/rail contact point is coupled to the various degrees of freedom of the motion; it is determined iteratively during the calculation.

Aerodynamic load
For a moving vehicle under the influence of a crosswind, the aerodynamic load on the vehicle is mainly caused by the high-speed airflow opposite to the forward direction of the vehicle and the ambient airflow. The aerodynamic loads are proportional to the square of the modulus of the sum of the two velocity vectors. The aerodynamic loads are given as: In Equations (11), ρ, A and L are the air density, reference area and reference length, respectively. The quantities C C , C L , C Mx , C My and C Mz are the coefficients of the lateral aerodynamic force, lift force, roll moment, pitch moment and yaw moment, respectively.
The vectors v and U are the reverse air velocity (i.e. in the direction opposite to the vehicle velocity) and the crosswind velocity, respectively. The vector U can be in any direction, although, in practice, it is most commonly in the crosswind sense. If a train travels at a speed of 300 km/h, the reverse air speed is 83.3 m/s. Under normal operating conditions, the reverse air speed is much greater than the crosswind speed. Therefore, the reverse air flow plays a major role in determining the aerodynamic load. The existence of a steady aerodynamic load has a major influence on the normal force on the wheel-rail system, because aerodynamic lift reduces the axle weight. The lateral aerodynamic force and the roll moment cause the normal forces on the left and right wheels to be different, and the pitch moment causes the normal force at the front wheel/rail contact point to be different from that at the rear wheel/rail contact point. The gravitational restoring force arises from the normal force between the wheel and the rail; a steady aerodynamic load thus changes the normal force and consequently changes the gravitational restoring force as well. In addition, the creep coefficient Equation (4) and correction factor Equation (8) are related to the normal force and the parameters of the contact point. Therefore, a steady aerodynamic load changes not only the normal force but also the equilibrium position of the wheel and rail. As a result, the creep coefficient and correction factor are changed significantly by the aerodynamic load and consequently so are the creep force and creep moment.
The changes in the gravitational restoring force and the creep force (and moment) due to a steady aerodynamic load appear in the equations of motion of the vehicle as changes in the stiffness and damping. This implies that the intrinsic characteristics of the vibration system of a high-speed railway vehicle also change. In other words, the effects of aerodynamic loads on a vehicle's vibration system are inherent and complex and are not just an external excitation.

Determining the non-linear critical speed
Because of the curved track and the aerodynamic load factors, the limit-cycle motion of a vehicle is asymmetric with respect to the centre of the track. In this paper, we construct the bifurcation diagram by plotting half of the peak-to-peak value of the lateral hunting motion as a function of vehicle speed. Figure 2 is a subcritical bifurcation diagram for a typical non-linear railway-vehicle system. The minimum bifurcation speed v n is the non-linear critical speed, and the larger bifurcation speed v h is the Hopf bifurcation speed. A correct method for calculating the bifurcation diagram and finding the minimum bifurcation point is based on the pathfollowing method [2]. In this method, we change the vehicle speed manually by a small step and solve the initial-value problem to obtain the dynamic response at each step. If the solution for the current step is stationary, then we increase the speed in the next step, and obtain the initial condition for the subsequent calculation by adding a small disturbance. The speed at which a periodic solution occurs is called the Hopf bifurcation speed v h . The solution for the current step is then taken as the initial condition for the next step. We continue increasing the speed until the amplitude of the periodic solution is significantly large. We then decrease the vehicle speed by a small step. The smallest bifurcation point, i.e. the non-linear critical speed v n , is found where the solution has a stable zero, as shown in Figure 2.

Simulation programme and validation
It is difficult to express the contact parameters as specific functions of the lateral displacement of the wheel relative to the rail. We therefore compute the wheel-rail profile at discrete data points and calculate the contact parameters using spline interpolation. We use the Runge-Kutta method to integrate the governing differential equations numerically to obtain the dynamic response, and we then calculate the bifurcation diagram, nonlinear critical speed and motion phase diagram. To ensure the reliability of the simulation programme, we compared the simulation results with those obtained from commercial software and from the literature.
First, we examined the submodule for the wheel-rail contact relationship. Figure 3 compares the results calculated by this submodule with those obtained from commercial software for a LMA/UIC60 pair and from Ref. [29]. The results compare well, indicating that the submodule is correct.   In Figure 4, we compare the lateral wheel/rail forces on the outer track of a railway vehicle during steady-state motion on a circular track. The agreement is good for all three velocity conditions.
Fortin [30] calculated the lateral displacements of wheelsets during negotiation from a straight track onto a circular track for both an ordinary railway vehicle and a vehicle with a turning mechanism. We simulated the lateral displacement curves of the wheelsets using our simulation programme, and subsequently compared the results with Fortin's calculations, as shown in Figure 5. The results are consistent, thus again validating the accuracy of the programme developed in this study.

Non-linear bifurcation analysis
On the basis of our dynamic model and the simulation programme described above, we calculated the bifurcation diagram and the non-linear critical speed for a railway vehicle on a curved track. We assumed the radius of the true circular curve to be 6000 m and the superelevation of the outer track to be 0.06 m. In the following discussion, we focus on two types of aerodynamic conditions: aerodynamic lift alone and crosswind loads.

Non-linear bifurcation analysis under aerodynamic lift
For aerodynamic lift, there are two cases: positive aerodynamic lift and negative aerodynamic lift. In our analysis, the positive aerodynamic lift coefficient is C L = 0.11563, and the negative aerodynamic lift coefficient is C L = −0.12441. When the effects of aerodynamic lift is not taken into account, C L = 0. Figure 6 shows bifurcation diagrams for the lateral movement of the first wheelset for a vehicle with positive aerodynamic lift, negative aerodynamic lift, and without any aerodynamic effect. In comparison to the case without any aerodynamic load, a positive aerodynamic lift increases the amplitude of the limit cycle and decreases the Hopf bifurcation speed and the non-linear critical speed. Conversely, a negative aerodynamic lift decreases the limit-cycle amplitude and increases the Hopf bifurcation speed and the nonlinear critical speed. See Table 1 for details. Thus, a negative aerodynamic lift improves the stability of a railway vehicle on a curved track, whereas a positive aerodynamic lift degrades its stability. Figure 7 shows the time history and phase-plane projections of the lateral displacement and yaw angle for the first wheelset at a speed of 290 km/h. The change in direction of the aerodynamic lift has a greater influence on the yaw than on the lateral displacement.

Non-linear bifurcation analysis under crosswind load
In this section, we consider further the complicated aerodynamic loads generated by crosswinds. On account of the curved track, we need to consider crosswinds in two opposite directions. One is a centrifugal crosswind, where the wind blows radially towards the outer rail of the track; the other is a centripetal crosswind, where the wind blows radially towards the inner rail of the track. To consider various wind directions and wind speed, we used the following five aerodynamic conditions in our calculations and analyses: (1) Aerodynamic effects not considered: vf = 0.0 m /s. We obtained the aerodynamic coefficients for the various aerodynamic conditions used in this section from Yang et al. [31]. These coefficients were calculated by the CFD method [15]. Table 2 shows all the aerodynamic coefficients. The difference between centrifugal and centripetal crosswinds is reflected in the opposite signs of C C , C Mx and C Mz .   We calculated the bifurcation diagram for a railway vehicle on a curved track for each of the five aerodynamic conditions listed above. Figure 8 shows the bifurcation diagrams for the lateral movement of the first wheelset and the third wheelset. These results show that the bifurcation diagram for a railway vehicle on a curved track has only one non-zero stable branch when considering the aerodynamic loads generated by crosswinds, which is the same as when the aerodynamic load effect is not considered. When the wind blows in the centripetal direction, the Hopf bifurcation speed and the non-linear critical speed increase. On the contrary, the Hopf bifurcation speed and the non-linear critical speed decrease under a centrifugal crosswind load. At a wind speed of 10.7 m/s, the changes in both the critical speed and the Hopf bifurcation speed are greater than those at a wind speed of 5.4 m/s. Table 3 shows the Hopf bifurcation speed and the non-linear critical speed for these five cases. The variations relative to the case without an aerodynamic load are shown in Table 3 as well. For a centrifugal crosswind at a speed of 10.7 m/s, the nonlinear critical speed is 6.4% smaller than that without an aerodynamic load. Conversely, for  a centripetal crosswind at a speed of 10.7 m/s, the non-linear critical speed is 12% larger than that without an aerodynamic load.
Generally, for a fixed vehicle speed, a centrifugal crosswind load increases the amplitude of the limit cycle, whereas a centripetal crosswind decreases it. Figure 8 shows that a crosswind load of 10.7 m/s has a significant effect on the bifurcation diagram. If the wind direction is centrifugal, bifurcation exists in a small range of speeds (218-225 m/s) near the critical speed. Compared with a centrifugal wind load, the limit-cycle amplitude of the third wheelset decreases sharply when a centripetal wind load is considered Figure 9 shows the time history and phase-plane projections of a vehicle moving at a speed of 280 km/h under a centrifugal crosswind load at v f = 10.7 m/s. The results show that-when considering a crosswind blowing in the centrifugal direction-the amplitude of vibration of the front bogie (including the first and second wheelsets and the front frame) is smaller than that of the rear bogie (including the third and fourth wheelsets and the rear frame). Figure 10 shows the time history and phase-plane projections of a vehicle moving at a speed of 280 km/h and under a centripetal crosswind load with v f = − 10.7 m/s. In a centripetal crosswind, the vibration amplitude of the front bogie is larger than that of the rear bogie. This is opposite to the case of a centrifugal crosswind load. This shows that a change in the wind direction can influence not only the non-linear critical speed but also the relative magnitude of the hunting motions of the front and rear bogies.

Frequency-spectrum analysis
To investigate the influence of aerodynamic loads on the vibration frequency of hunting motions, we used the fast fourier transform algorithm (FFT) to perform frequencyspectrum analyses of the wheelset motions. The sampling period was equal to the numerical integration step, 0.001 s. Thus, the sampling frequency was 1000 Hz, which meets the accuracy requirement of the FFT. Figure 11 shows the frequency spectrum of the first wheelset when limit-cycle motion occurs at a vehicle speed of 290 km/h, with aerodynamic  lifts included. The frequency spectrum at the non-linear critical speed under a crosswind load is shown in Figure 12. When hunting motion occurs, multiple frequencies appear in addition to the fundamental frequency, including the double frequency, triple frequency, quadruple frequency and higher frequencies.
Unlike a linear vehicle system, in a non-linear vehicle system, higher frequency vibrations occur at multiples of the fundamental frequency. In this section, we analyse the cause of the higher frequency vibrations for a vehicle subject to an aerodynamic load on a curved track. The non-linearity of our dynamic vehicle model is mainly rooted in the non-linear wheel/rail contacts. For simplicity, we analyse below the lateral movement of a wheelset. The relevant equation can be written in the simplified form where F k and F f are the primary-suspension force and the wheel/rail creep force, N L and N R are the left-and right-contact normal forces and λ L and λ R are the left and right contact angles, respectively. We consider the LMA/UIC60 wheel/rail pair as an example. The relation between the wheel/rail contact angle and the lateral displacement y w of the wheelset can be fitted with a polynomial function, as shown in Equations (13) and (14) and Figure 13. In Figure 13, y w = 0 means that the wheelset is located at the centre of the track and that the wheel/rail contact point is located on the nominal rolling circle.
When a vehicle is running on a straight, flat track without any aerodynamic load, its equilibrium position is located at the centre of the track, and the corresponding wheel/rail contact point is located on the nominal rolling circle. Then, the relationships between the contact angles and the relative lateral displacements of the left and right wheel/rail pairs are symmetrical, as shown in Figure 13. In the neighbourhood of the equilibrium position, the normal contact forces also can be regarded as equivalent; i.e. N L = N R . Substituting Equations (13) and (14) into Equation (12) and subtracting N R ·λ R from N L ·λ L causes the even terms in the polynomial to disappear, and only odd terms remain. However, if the vehicle is running on a curved track or under a crosswind load, its equilibrium position deviates from the centre of the track. Naturally, the normal contact forces in the neighbourhood of the equilibrium position then cannot be regarded as equivalent, and the relations between the contact angles and the relative lateral displacements of the wheel/rail pairs become unsymmetrical. In this case, both even and odd terms remain in the polynomial after N R ·λ R is subtracted from N L ·λ L . From this analysis, we conclude that only higher frequencies that are odd multiples of the fundamental frequency appear in the hunting motion when the equilibrium position is located at the centre of the track. Nevertheless, if the equilibrium position deviates from the track centre, then higher frequencies that are both even and odd multiples of the fundamental frequency appear in the hunting motion. The deviation from the equilibrium position may be caused either by the curved track or by the crosswind load. Although we obtained this conclusion by analysing the specific LMA/UIC60 pair, it can be extended to other wheel/rail pairs because the relationships between the contact angles and the relative lateral displacements of the left and right wheel/rail pairs are always symmetrical when the vehicle equilibrium position is located at the track centre. To validate this conclusion, we simulated the hunting motion of a vehicle on straight track for cases both with and without aerodynamic loads, and we analysed the frequency spectrum. As shown in Figure 14, for a vehicle on a straight track and without an aerodynamic load, the equilibrium position is located at the centre of the track, and the high-frequency vibrations only contain frequencies that are odd multiples of the fundamental frequency. However, the equilibrium position deviates from the track centre when there is a crosswind load, and it is obvious that frequencies that are both even and odd multiples of the fundamental frequency exist in the hunting motion. These simulation results thus confirm our inference.
Our previous work [22][23][24][25] on the linear stability of a railway vehicle was performed using eigenvalue analysis, and we determined the influence of an aerodynamic load on the linear critical speed. In this paper, we give an explanation using explicit mathematical expression for the phenomenon that higher frequencies are multiples of fundamental frequency under actions of crosswind in nonlinear system.

Conclusion
A steady aerodynamic load changes the normal force at the various wheel-rail contact points, thus changing the creep force, gravitational restoring force and moment. This also changes the equilibrium position of the railway vehicle on the curved track. These aerodynamic effects change the stability of high-speed railway vehicles on a curved track. To investigate the effect of aerodynamic loads on the hunting stability of a vehicle on a curved track, we simulated the limit-cycle motions and bifurcation diagrams and obtained the critical speed for such a system. We also performed frequency-spectrum analyses and briefly discussed the mechanism responsible for the high-frequency characteristics of the hunting motion. From these analyses, we have drawn the following conclusions: For a railway vehicle running on a curved track subject to aerodynamic loads, when hunting motions occur, multiple frequencies-including the double frequency, triple frequency, quadruple frequency and higher frequencies-as well as the fundamental frequency appear. In addition, a curved track or a crosswind load causes the equilibrium position to deviate from the track centre. In such cases, higher frequencies that are both even and odd multiples of the fundamental frequency appear in the hunting motion.
However, when the equilibrium position is located at the centre of the track, only higher frequencies that are odd multiples of the fundamental frequency appear.
A negative aerodynamic lift can improve the non-linear stability of a vehicle, whereas a positive aerodynamic lift can degrade the stability. Compared to a case without any aerodynamic load, a negative aerodynamic lift reduces the limit-cycle vibration amplitude of the various degrees of freedom of the railway vehicle. The greater the vehicle speed, the more pronounced the effect. Conversely, a positive aerodynamic lift increases the limit-cycle vibration amplitude.
A centrifugal crosswind load can lower the non-linear critical speed and the Hopf bifurcation speed of the vehicle. Conversely, a centripetal crosswind load can increase the non-linear critical speed and the Hopf speed. In one calculation, we assumed the radius of the curve to be 6000 m, the outer rail superelevation to be 0.06 m and the crosswind speed to be 10.7 m/s. For a centrifugal wind direction, the Hopf bifurcation speed is then reduced by 4.4%, and the non-linear critical speed is reduced by 6.4%. For a centripetal wind direction, on the contrary, the Hopf bifurcation speed is increased by 11.0%, and the non-linear critical speed is increased by 12.0%.
A change in the crosswind direction changes not only the non-linear critical speed but also the relative amplitude of the hunting motion of the front and rear bogies. At a wind speed of 10.7 m/s and a centrifugal wind direction, the hunting amplitude of the front bogie is smaller than that of the rear one. However, when the wind direction is centripetal, the hunting amplitude of the front bogie actually becomes greater than that of the rear bogie.

Disclosure statement
No potential conflict of interest was reported by the authors.