ANALYSIS OF THE CONDITIONS FOR THE EXHAUSTION OF THE STABILITY MARGIN IN THE RAIL TRACK OF FREIGHT CARS WITH THREE-PIECE BOGIES

The research on improvement of methodical approaches to definition of the probable reasons of infringement of conditions of stability of freight cars from derailment is carried out. Using a basic computer model of the dynamics of a freight car, the influence of the characteristics of the technical condition of their running gear and track on the indicators of empty cars stability from derailment was studied through the computational experiment. The article presents the main statements of the research methodology, which provides the analysis of probable causes of derailment of freight cars by conducting a series of numerical experiments with logging the progress of calculations and saving the results. Factor analysis was used to interpret the calculated data with an assessment of each of the factors influence or their combination on the probability of derailment. The developed procedure of the simulation experiment provides a step-by-step study of the freight cars derailment conditions, including factors structuring and ranking, development of experimental plan, calculating coefficients of wheel pairs resistance to derailment from rails, provided that the wheel flange rolls onto the rail head, and determining the degree of influence of relevant factors on the dynamic stability of cars from derailment. A comparative analysis of the stability of cars in rail tracks was performed using the introduced concept of the combined coefficient of stability of wheel pairs against derailment. Determining the probable causes of car derailment is based on scanning the parameter field. The results of the parametric study revealed the degree of influence on the freight cars stability of running gear technical condition characteristics. In particular, it is determined that the most dangerous in terms of stability loss of empty cars in the track is the exceeding of the wedges of the vibration dampers.


Introduction
Railway accidents involving the derailment of rolling stock depend on many factors, both objective and subjective. Due to the combined action of many factors, some of which are not recorded by objective means of control during the movement of the train, the analysis of emergency situations is not always possible to identify and explain the cause of the derailment. At the same time, the assessment of traffic safety indicators according to existing methods does not reflect the actual conditions that increase the risks of derailment. Railway safety as a key issue includes a wide range of components, among which a leading place belongs to the dynamics of vehicle motion ( A characteristic feature of the spatial oscillations of vehicles, the movement of which is directed by the rail track, is the tendency under certain conditions to self-excitation of auto-oscillations (Mazilu, T., 2009). The lowest value of the speed at which unquenchable lateral oscillations of a railway vehicle occur, is called the critical speed of hunting Vcr. Transverse hunting oscillations of bogies when moving at speeds exceeding the critical, cause excessive lateral forces of the wheels on the track, increase the damage of freights sensitive to dynamic loads, and lead to additional damage to rolling stock and railway infrastructure. In addition, the extra energy of the locomotive is spent on maintaining a constant speed of the train composed of cars, the movement of which is complemented with self-oscillations. Thus, the critical hunting velocities determine the limits of threshold changes in the dynamic properties of railway vehicles.
Self-excitation of lateral oscillations of railway vehicles is caused by their loss of stability of undisturbed motion. Effective use of methods of mechanical stability theory in relation to studies of the dynamics of the movement of railway vehicles, in particular the first approximation of A.M. Lyapunov (Lyapunov, A.M., 1956), first carried out by academician V.A. Lazaryan (Lazaryan, V.A., 1964).

Dynamic phenomena contributing to emer-
gencies related to rolling stock derailment occurrence Obviously, each transport event is associated with a coincidence of a number of adverse circumstances, among which, however, there is always a leading cause. The low freight cars stability margin from derailment is most often caused by their unsatisfactory dynamic properties which are mainly explained by design features and a technical condition of running gear (Galiev, I.I., et al., 2011). According to the results of numerous studies and investigations of transport accidents, it turns out that the objects of emergency situations are increasingly freight cars in an empty state (Ge, X., et al., 2018; Ermakov, V.M., Pevzner, V.O., 2002). Empty cars with a high center of mass (bunker cars, tank cars, etc.) are most prone to loss of stability in the rail track.

Resonant movement modes of freight car in empty state
It is known that the spring suspension of freight cars of 1520 mm gauge in the empty state can partially or completely lose its damping properties due to the weakening or complete exclusion from the operation of wedge vibration dampers caused by the so-called exceeding of the wedges. At the same time rigidity of a spring suspension bracket of the bogie can decrease in 1.4 times. This situation leads to a decrease in the natural frequencies of the car, and hence a decrease in the speed at which the resonant mode occurs. Periodic perturbations that cause resonant modes of rolling stock are associated with both periodic irregularities of the track and the existing defects on the rolling surfaces of car wheels. Therefore, in the spectrum of perturbations acting on a moving car, there are always components with the frequency of rotation of the wheel pairs. These components, even with the permissible defects of the wheels, are sufficient for the development of resonant phenomena, when the speed of the car reaches a critical value when the wheel pair rotation speed fw with one of the natural frequencies fi . These velocities are called resonant. So the resonant speed Vr is calculated by expression Vr = Lw · fi , where Lwis the length of the wheel rolling circle. Fig. 1 shows a diagram for determining the resonant velocities. Here, rays I and II show the dependences of the speed of wheel pairs with the full (I) and limiting (II) thickness of the wheel rims. Resonant velocities Vr (i) (i = 1…4) are at the points of intersection of lines I and II with the horizontal natural frequencies of vertical oscillations fv and fv * respectively at nominal and reduced (due to the exclusion of underwedge springs) stiffness of the suspension. Thus, under operating conditions, the resonant velocities can take values in the range from Vr (1) to Vr (4) . The most dangerous for empty cars is the Vr (1) -Vr (2) speed range, which corresponds to cases of insufficient or absent damping of oscillations (Fig. 1). In the resonant modes of oscillations of bouncing and pitching at the moments of full unloading of wheels in case of horizontal forces cross there is a real threat of derailment. The natural frequencies of oscillations of the freight car bodies of some types in the empty state are shown in Table 1. Oscillation frequencies of bouncing fb and fb * , pitching fp and fp * and rolling fr and fr * calculated according to two values of the stiffness of the spring suspension, corresponding to the nominal stiffness of the spring sets (numerator) and reduced due to the exclusion from the normal operating state of the wedge vibration dampers (denominator). According to the calculated frequencies, the resonant speeds of the considered types of cars with new and worn wheels are determined. For example, Fig.  2 shows the range of resonant velocities of the empty gondola car at full and maximum thickness of the wheel rim, respectively, at nominal and reduced stiffness of the suspension, taking into account the calculated frequencies (Table 1). Lines 1, 2, 3 correspond to frequencies fb, fp and fr and lines 4, 5, 6to frequencies fb * , fp * and fr * . Dependences of frequencies of rotations of wheelsets on full ff(v) and limit fut(v) the wheel rims thicknesses that corresponding to the wheel radii of 0.475 and 0.425 m are represented by graphs 7 and 8. Due to the presence of open pairs of dry friction in the combinations of bearing elements of the running gear between themselves and the body, it is possible to stop in the relative movements of individual bodies of the system, which includes the model of the freight car. Thus, the system may lose degrees of freedom and move from one structural state to another. Therefore, the original design system of the car can be considered as a system with a variable structure. The number of possible structural states of such system is equal to 2 i (inumber of friction nodes). Based on the concept of fundamental variability of the output system, which simulates the dynamic behavior of a freight car, a method for determining critical velocities using linearization of discrete systems with dry friction units was proposed (Diomin, Yu.V., et al., 1994). The essence of this method is to replace the original nonlinear system with l linear subsystems (l = 2 i ). Each of l subsystems corresponds to one of the possible states of the original nonlinear system. Such subsystems are built in accordance with the structural changes of the original system due to the alternate closure of connections with dry friction. When constructing linear subsystems, the main thing is to determine the parameters of viscous friction, which replaces dry friction in open joints. According to the developed method of formation of linear subsystems for determining the coefficient of equivalent viscous resistance in connection of a multi-mass self-oscillating system is carried out according to a formula similar to that used by S.P. Tymoshenko in the study of forced oscillations of the oscillator with dry friction (Weaver, W. Jr., et al., 1990). Regarding the model of operation of the body support devices on the bogies during their mutual turns, the mentioned formula has the form 11 where: The indicators of stability of the least stable of a number of subsystems, which approximated the original system, determine the conditions of selfoscillations of the studied railway vehicle. Thus, the method of structural linearization allows to extend powerful methods of linear algebra to a class of systems that are not fundamentally linearizable.

Estimation of stability margin of wheel sets
from derailment A necessary step in determining the prerequisites for derailment is to study the influence of certain factors on the characteristics of the dynamic processes that accompany the movement of the car. The assessment of dynamic characteristics should be the selected indicators of the margin of stability of wheel sets from derailment under the condition of rolling the wheel flange on the rail head, which comprehensively characterize the combination of both horizontal and vertical forces acting simultaneously on the wheel of each wheel set. On 1520 mm gauge railways, the main indicator of rolling stock safety is the so-called coefficient of stability of wheel set from derailment, provided that the wheel flange rolls onto the rail head. (Standards, 1996). Coefficient of stability of wheel against derailment when moving the car with the maximum speed on the straight track of good condition with combinations of deviations in the plan, skews and sags allowed, is calculated by the formula: − determining the degree of relevant factors influence on the dynamic stability of cars from derailment. The probable causes of car derailing are determined on the basis of computer simulations based on a scan of the parameters field. This method provides complete information about the objective function within the defined parameter sets. The number of computer experiments when scanning is calculated as N = m k , where k is number of varying factors, m is the number of levels at which each factor varies. Depending on the number of factors and the levels of each of them selected for the scan, the number of options is growing rapidly. Thus, the time of scanning and computational costs, increases significantly. In the task for scanning, in addition to the speed V, as study factors the following characteristics of the undercarriage were selected: fpfriction coefficients in center bearing nodes; wpdislocation of center bearing nodes in the longitudinal direction; fsfriction coefficients in the side slides; klexceeding of wedges of bogies; wb1 and wb2 -clearances in the longitudinal direction between the axle boxes and side frames, respectively, for the first and the second bogies. For each of the selected factors, the determined levels are shown in Table 2.  The full-factorial plan of the experiment as a plan of the conducted experiments takes into accounts all possible combinations of levels of each factor. According to the identified factors, a full-factorial plan of the experiment with the total number of variants 144 was formed (Table 3). An experiment with such a plan allows us to quantify the effects of both individual factors and the interaction of factors (Adler, Yu., et al, 1971).   According to these options, the dynamics of the gondola car in the empty state were calculated by computer simulation. The comparative analysis of the received results is carried out on the combined indicator of stability from derailing of the gondola car kdr0, which was calculated as the smallest of the minimum values derailment stability margin coefficients kdr1, kdr2, kdr3, kdr4 respectively for each wheelset (min min). Fig. 3 shows the values of the combined coefficient of resistance to derailing kdr0 at speeds of the gondola 60, 70, 80 and 90 km/h on a straight track section of a satisfactory condition.
At speeds of 60 and 70 km/h, all calculated values of the coefficient kdr0 are higher than the allowable level. However, the margin of stability significantly depends on changes in parameters and characteristics of the technical condition of the running gears. For example, as obtained in the calculation variants 82-90 and 100-108, the exceeding of the wedges worsens the situation with derailing of the gondola only in combination with excessive friction in both central bearing nodes and on side bearings.

Stability of cars of different types in a rail track
Calculations for gondola car, covered car, hopper car and tank car were performed to determine the effect on the stability of the freight car type. In this case, given the fact that the clearances in the longitudinal direction between the axle boxes and side frames affected the resistance to the derailment of the empty gondola less than other changes in the technical condition, only 16 options were considered calculating the numbers N = 5 + 9i, where i = 1…15 (Table 3). Fig. 4 and 5 show the values of the combined coefficients of stability kdr0 for four types of freight cars in the empty state at speeds of 60 and 80 km/h, respectively. Here the lines connecting the calculated values kdr0 are marked as follows: 1for gondola car; 2for covered car; 3for the hopper car; 4for the tank car.