The papers where problems of railway vehicles’ stability and determination of nonlinear critical velocity v_n are discussed by present authors are [3,6,7,8]. A broad reference to the corresponding literature by other authors is provided in these publications, too. The newest example papers on the stability of rail vehicles and their dynamics, also with account taken of motion in CC and TC, are [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. In [9], its authors study the stability of high-speed trains through a focus on bogie behaviour. The novel method of stability determination, based on the root loci methods, is successfully tested. A sensitivity analysis was performed on the selected vehicle parameters, too. In [10], one can see the interesting differences in stability issues for rail and road vehicles. The important influence on the stability of vehicle roll angle is considered. The results for the stability region are presented in a 3-dimensional stability domain. Paper [11] describes a novel measuring system that monitors vehicle hunting motion. The system can predict the wheelset’s lateral and yaw displacements and wheel-rail contact relationships in real time based on the monitoring results. The accuracy and efficiency of the whole system were validated by comparing the predicted results with simulated and experimental results. Paper [12] is truly interesting and of high cognitive value. It reveals and verifies experimentally (on a roller rig) the possible existence of parametric vibrations in the wheelset-track system. They appear parallel to self-exciting vibrations (hunting motion) and are caused by wheel load fluctuations. The systems particularly prone to such type of resonance are those with big tread angles. These last also decrease the critical (hunting) velocity, which favours simultaneous existences of self-exciting and parametric vibrations. Publication [13] represents the study of the chaos in a mechanical impacting system. The novelty of this paper lies in the extension of previous results for a 1 degree of freedom (DoF) system on a 2 DoFs system. The existence of phenomena such as narrow band chaos, finger-shaped attractors etc., was demonstrated numerically and then experimentally verified. In [14], the authors propose a new method for stability determination based on the data instead of the equations. With this method, they managed to determine long-term statistics of the chaotic state, the covariant Lyapunov vector, the Lyapunov spectrum, the finite-time Lyapunov exponents, and the angles between the stable, neutral, and unstable splitting of the tangent space. Publication [15] presents and uses one more nonlinear wheelset model to perform bifurcation analysis for him. The nonlinearity in the wheel-rail contact is of primary interest. The analysis comprises comparisons between results for the linearised and nonlinear models. The results for different field-measured wheel profiles are also compared. Finally, the conclusion is that larger suspension stiffness would increase the running stability under wheel wear. Publication [16] is to some extent similar to [15]. The nonlinear wheelset model undergoes the bifurcation analysis. The nonlinear equivalent conicity in wheel-rail contact is of interest. Results for the linear and nonlinear approaches to equivalent conicity and contact forces are compared. The authors show the possible coexistence of stable and unstable limit cycles and expand on the consequences of Chinese high-speed trains' observed unfavourable behaviour. In the paper [17], the authors study the influence of curved track parameters, such as the radius, superelevation, TC and CC lengths, on vehicle-track interactions in the case of side-frame cross-braced 2-axle railway bogie. They conclude that curve radius is of unequivocal importance, TC length is important provided the inflexion point of this importance is not exceeded, superelevation is of minor importance (differences in results for its deficiency and excess are surprisingly small), and CC length is of no importance at all. In [18], the authors focus on vibrations in passenger cars. They compare results from two approaches, one based on analytical solutions of the dynamical equations and the other based on simulation, which means solving the equations numerically. Thanks to the bigger possible DoF’s number and smaller simplification of the equations for the simulation approach, the authors conclude that it is superior to the other approach. The authors highlight the higher accuracy of the simulations. Publication [19] compares a few methods of nonlinear critical velocity determination. The simulation approach to this issue and the ramping and path following (continuation) methods are of primary interest. Different methods of hunting motion excitation are compared with each other in terms of the critical velocity value. The influence of the track irregularities (track class) on the results is shown, too, thanks to comparison with the results for an ideal track. In [20], the influence of the bogie suspension parameters on the track frame lateral forces is studied. The bogie is a three-piece traditional construction applied in freight cars. The authors consider 30 different bolster suspension combinations. It is concluded that increasing the dumping force (the wedge coefficient of friction) increases the lateral frame forces. Simultaneous increases in the lateral stiffness of the bogie and the dumping force increase the lateral frame force, too. On the other hand, the internal interplay between these two parameters appears also important. In [21], the hunting of locomotive carbody is studied experimentally and with simulation. The aim is to explain and suggest measures to eliminate such hunting that appears for low conicity in wheel-rail contact. The measures concern the suspension parameters. Namely, the series stiffness and damping in the yaw damper and the longitudinal stiffness in primary suspension are indicated as those needing the decrease. Both the ST and CC cases have been analysed. In the paper [22], the author studied interactions between vehicle internal elements and between vehicle and track to make sure that existing freight cars with three-piece bogies can run at higher permissible speeds under the increased axle load. The ST and CC sections are considered. It is shown that such increases are possible. However, the bogies have to be replaced with slightly modernised ones. In [23], the authors propose a method to identify, rather than neglect, small amplitude hunting of high-speed railway vehicles. They found that the autocorrelation coefficient and spectral frequency spread are the most efficient parameters for this purpose. The findings are dedicated to supporting the monitoring of hunting instability and the real-time active control studies of high-speed trains. Paper [24] is a general paper on the instability of the limit cycle type oscillations. The aim is to address the need for an efficient computational approach to model instability and handle hundreds or thousands of design variables. It is fulfilled by using a simple metric to determine the stability of the limit cycle utilising a fitted bifurcation diagram slope. The stability derivative of many design variables is efficiently computed with the developed adjoint-based formula. Publication [25] proposes the wheel-rail nonlinear kinematics model, which is an extension of the well-known linear model by Klingel. The nonlinear model comprises high-order odd harmonic frequencies (HOHFs), apart from the fundamental frequency of hunting. Like this last, the HOHFs make the self-exciting vibrations source in rail vehicles. Thus, these findings are important for the methods of hunting instability research and condition monitoring of trains.

Summarising the content of these publications, one can note that their scope is similar to that of the profiled literature from 15 years back. So, we can find the works of cognitive character, the works on new methods of research and description, the works on general dynamics, stability issues, hunting (limit cycles), chaos and so on. Still, the number of works concerning issues for ST is superior to those for CC and TC problems.

Although part of the discussed literature directly concerns the stability issue, the fundamentals of the stability phenomenon are given below.

The physical explanation for practical problems of rail vehicle stability is the self-exciting vibrations that appear in vehicle-track systems. They are a phenomenon known for a long time [26]. Some vehicle parts start to undergo self-exciting vibrations at certain motion conditions (parameters). The wheelsets oscillate laterally and around the vertical axis (yawing). The flange wheel-rail contact, noise, and higher risk of derailment accompany them. Although the main vehicle motion along the track is usually possible in such conditions, the self-exciting vibrations make evident adverse phenomena in rail vehicle operation in straight track (ST) and circular curve (CC) sections. The self-exciting vibrations observed in real vehicle-track systems [26] can be identified using the vehicle-track model with bifurcation of its solutions [27,28]. Bifurcation analysis is an effective method to test the modelled object [27,28,29,30] and is thus applied also in the presented research. The numerical simulation studies of the vehicle-track system enable us to observe if the model solutions are stable. The stable stationary solutions (constant value of the model’s observed coordinate in ST and CC) describe the normal exploitation conditions of a real vehicle. It means that in the real system, each disturbance, for example, caused by track irregularity, is effectively damped while the system tends to equilibrium. The properly designed vehicle should ensure running stability in the expected range of exploitation parameters changes, such as velocity (0 to v_max), load (empty, partially to fully loaded), load asymmetry, track shape (CC, transition curve (TC), ST), track maintenance, etc. The stable stationary solutions of the modelled vehicle-track system in ST and CC are expected for typical operation conditions. In contrast, due to constant change of curve radius and superelevation in TC, stationary solutions in TCs are neither possible nor expected.

The model's stable periodic solutions (limit cycles) are attributed to self-exciting vibrations in the real system. For them, the lateral displacements of a wheelset change periodically during main vehicle motion (along the track). Calling such solutions “stable periodic” is justified because they are limited, observable and can last any length. On the other hand, stable periodic solutions represent the highest abilities of the modelled system in terms of stable behaviour and also bring other valuable information. Nonlinear critical velocity v_n is the key parameter in the stability analysis, and so is the presented research. The v_n value separates the range of vehicle velocity v, where stable stationary solutions exist (v < vn), from the range, where stable periodic solutions may appear (v ≥ v_n). The v_n value is the minimum vehicle velocity v, at which stable periodic solutions can appear. For a real object, it means self-exciting vibrations may occur, and the vibrational system’s state is reached. Further increase of system output (for example, with a rise in v) leads to an increase in the self-exciting vibration amplitude. Finally, amplitude values are not constant anymore, which means unbounded growth of the solutions (loss of stability), which could lead to an accident due to the derailment in the real system. The research of vehicle model periodic solutions for the increased system input (vehicle velocity) in the over-critical system state is interesting. Its conduction is done to reconnoitre how much the system parameters (velocity) can be exceeded to keep the solutions stable (periodic or stationary). A wide range of the system velocity between the value corresponding with periodic solutions' appearance and loss of stability is always desirable for the real vehicle.

The stability maps for the motion of the empty wagon model are shown in Figure 4. As can be seen there, the properties of the model are regular. That is, for each of the tested routes the critical velocities v_n were determined. Just stable stationary solutions appear for velocities lower than the critical ones. The periodic solutions exist for velocities equal to and greater than the critical values v_n. The values of critical velocity v_n on each curved track increase with the curve radius decrease (Table 2). The smallest value is 37.2 m/s for an ST (R = ∞), and the highest v_n is 45.2 m/s for a CC with a R = 1200 m radius. Stable periodic solutions still exist at velocities greater than 200 m/s in the ST (black line). The velocity ranges where periodic solutions in CCs occur are smaller than in ST and decrease as the curve radius decreases (different line colours for particular R values). For each CC section and the final range of velocities, there is a bifurcation of periodic to stationary solutions, which are the stable ones. Potentially, it could be the effect of a centrifugal force action. Only stable stationary solutions exist in the range of velocities lower than the critical value, with |y_lw|max values increasing according to the curve radius decrease. Therefore, the diagrams show results for motion velocity higher than 30 m/s. The lack of wheelsets' central location in CCs below v_n (|y_lw|max ≠ 0) results from motion conditions, including the applied track superelevation (Table 1).

So, in light of the tests carried out on the car model in an empty state, the requirements regarding motion stability for operating velocities (less than 120 km/h ≈ 33.33 m/s) are met. The critical velocities on particular tested routes are not lower than the maximum operational velocity. The periodic solutions with |y_lw|max values up to approximately 0.006 to 0.007 m appear for the critical- and bigger than critical velocities. This means that the wheelsets’ self-exciting vibrations may occur in the modelled system. Still, the basic vehicle motion (along the track) is possible in the over-critical velocity range, too.

The laden vehicle model presents significantly different features in the same motion conditions (track curvatures, Figure 5) as compared to the empty one. The lowest value of the critical velocity occurs on an ST, where v_n = 30.8 m/s. It is lower than the maximum vehicle operating velocity, i.e. 120 km/h ≈ 33.3 m/s. This means self-exciting vibrations may appear in the modelled car during regular operation. Critical velocities are higher in CCs (Table 2) than in ST. In the range of overcritical velocities, there exist bifurcations of solutions: from periodic to stationary and from stationary to periodic on an ST and CC of the R = 6000 m radius. These make the stability maps hardly legible. Therefore, the analysis of selected solutions on the routes for the loaded car is presented in separate diagrams further on.

A moderate effect of reducing the lateral stiffness in the primary suspension can be observed in motion along an ST (Figure 10). The critical velocity v_n decreases from 37.2 m/s for nominal k_zy value (38.9⋅10⁵ N/m), to 36.5 m/s for 90% ‧ k_zy ≈ 35⋅10⁵ N/m, and to 35.8 m/s for 80% ‧ k_zy ≈ 31⋅10⁵ N/m. So, the v_n is greater than the required minimum v_n = 33.3 m/s. The nature of the model solutions is analogous for each k_zy value. That is, for velocities lower than v_n, there are stationary solutions only, and for velocities greater than v_n, there are periodic solutions that persist up to velocities greater than 200 m/s. The values of the solutions are similar for each value of k_zy.

The influence of the k_zy on the investigated model parameters is also moderate in the case of motion in curved tracks. The critical velocity is 40.1 m/s for the nominal k_zy value and large radius curve of R = 6000 m (Figure 11). Reducing the lateral stiffness causes a slight decrease in the critical velocity to the values v_n = 39.6 m/s for 90% k_zy and v_n = 39.2 m/s for 80% k_zy, respectively. Stable periodic solutions persist for increased velocity up to about 151 m/s, followed by bifurcation to stationary solutions. Stable stationary solutions exist up to the velocity of about 176 m/s. The model solution values for particular k_zy applied, are similar in the entire velocity range.

The critical velocity is 44.4 m/s on a CC with a smaller R = 3000 m radius, for the nominal value of k_zy = 38.9⋅10⁵ N/m (Figure 12). Reducing the lateral stiffness to 90% k_zydecreased the critical velocity to 44.2 m/s. Further reduction of the lateral stiffness to 80% k_zy reduced the critical velocity to 44 m/s only. So, a moderate influence of the k_zy decrease on the critical velocity on this route is also observed. Values of the model solutions (|y_lw|max and p-t-p of y_lw) for individual k_zy are similar to values on the previously studied routes. Thus, it can be observed that reducing k_zy to 80% of the nominal value reduces the model's critical velocity in the empty state, but not more than by 1 m/s.