## 2 The Approach to Stability and the Corresponding Model
A pair of individual viscous dampers acting vertically \(c_{zz}\) and laterally \(c_{xy}\) are finally adopted to simplify the bogie model. Vertical coil springs (formed as two-spring sets) in each axle box represent the stiffness in the vertical \(k_{zz}\) and lateral \(k_{zy}\) directions. The parameters of stiffness and damping just mentioned take two values each. One refers to empty gondola conditions and the second to laden gondola conditions (see Table A in Appendix A). In this simple way, the dependence of the parameters on the vertical load is considered to some extent. The values of the stiffness and damping parameters were obtained following the identification procedure performed at the authors' home faculty [33]. Experiments on a real freight car with Y25 bogies were its basis. Shortly, for the damping, the measured car's responses enabled formulae for vertical and lateral damping forces to be built and then the linear equivalent coefficients for different discrete car load states in the indirect procedure to be determined. The result is somewhat simplified. Nevertheless, the procedure is multi-level and scientifically advanced. On the other hand, the literature shows that damping forces can be crucial to vehicle dynamics. An example is a recent vertical dynamics study [34] for high-speed rail where comfort issues for vehicles running up to 400 km/h on a track with vertical irregularities were tested. It confirmed that when considerable vertical dynamical interactions between the vehicle and the track exist and are of interest, the influence of the vertical load on the damping forces (dry friction dampers) should be as accurate as possible. The conditions in the present study are different, however. The vertical dynamics is out of scope, while the vehicle's admissible speed \(v_{ad}=120\) km/h. Equally, the track is perfect, which means no vertical or lateral irregularities are considered. Consequently, vertical load changes (influencing forces for the dry friction damper laterally and vertically) almost do not exist. Additionally, in such circumstances, the well-known and commonly applied assumption easily holds that the couplings between the rail vehicle's lateral and vertical dynamics are insignificant. Then, analyses can be carried out separately for the lateral and vertical directions, e.g., [10, 14, 15, 24, 33, 34]. In practice, no mutual influences of vertical and lateral dynamical solutions can be observed. Thus, the simplified modelling of vertical dumping does not influence the lateral dynamics (lateral stability) in the present study. The longitudinal guidance of the wheelsets is very stiff. It represents a type of horn guidance [28]. So, eventual small deflections in the guide's material only effect the longitudinal stiffness \(k_{zx}\). Because of this, it is agreed that \(k_{zx}\) is independent of the car load. The freight car model is supplemented with a flexible track model, both vertically and laterally. The track parameters correspond to a ballasted European track with a 1435 mm gauge. Nominal profiles of UIC60 rails with an inclination of 1:40 are adopted. Nominal profiles of S1002 wheels are taken. The parameters of nonlinear wheel-rail contact are calculated using the ArgeCare RSGEO software. The tangential forces in the wheel-rail contact are calculated using [PERSON]'s simplified theory, implemented in the FASTSIM numerical procedure [35, 36]. Integration of the equations of motion is performed using  
Figure 2: Nominal model of the 412W gondola–track system: (**a**) side view, (**b**) front view, (**c**) top view. Gear's procedure. The error and step size control characterise its features. Its suitability for solving stiffness problems is well known, as with rail vehicle dynamics issues [28, 29]. The whole model of the vehicle-track system possesses 76 kinematic DoFs. Appendix A (Table A1) comprises detailed parameter values for the whole model.