DETERMINATION OF THE ISSUE CONCERNING THE LIFT RESISTANCE FACTOR OF LIGHTWEIGHT CAR

EDSD MBCSS, Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnіpropetrovsk, Ukraine, 49010, tel. +38 (050) 214 14 19, e-mail angela_shvets@ua.fm, ORCID 0000-0002-8469-3902 EDSD MBCSS, Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnіpropetrovsk, Ukraine, 49010, tel. +38 (095) 545 38 87, e-mail constantinz@i.ua, ORCID 0000-0003-3648-1769 EDSD MBCSS, Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnіpropetrovsk, Ukraine, 49010, tel. +38 (067) 178 16 90, e-mail asakulov@gmail.com, ORCID 0000-0002-6123-5431 EDSD MBCSS, Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnіpropetrovsk, Ukraine, 49010, tel. +38 (067) 282 13 41, e-mail zabolotnyi@i.ua, ORCID 0000-0003-1651-7082 EDSD MBCSS, Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnіpropetrovsk, Ukraine, 49010, tel. +38 (066) 633 55 95, e-mail 457m@ukr.net, ORCID 0000-0001-5695-5955


Introduction
Traffic safety is a basic condition for normal operation of the railways.Problems of traffic safety of trains and shunting operations are the main ones for the rail transport since it is hard to completely avoid crashes and accidents occurring due to car derailment [9].
The main causes of accidents in locomotive economy of railways of Ukraine are a combination of car malfunctions and track condition, as well as foot-plate staff's actions leading to violations of train driving modes.
The traffic accident analysis of the locomotive facilities of Ukrainian railways for the period of 2008 -2010 (Fig. 1) shows that most of accidents are caused by the actions of foot-plate staff (Fig. 1a) and freight traffic -freight electric locomotives and diesel locomotives (Fig. 1 b) [1,16,20].
Since the driving of freight trains results in quite large longitudinal forces acting on cars, it becomes important to conduct the research aimed at identifying the causes of the situations in which the cars might be lifted.Increased safety of freight car traffic is one of the priorities of the Ukrainian railways and it represents a set of measures aimed at reducing the likelihood of threat to life and health of passengers, transported goods, as well as at the safety of infrastructure and rolling stock of the railway transport, environmental safety.
For effective training of specialists in such a complex area as railway transport currently the simulator complexes are widely used [10].The simulator is usually used for developing practical skills, bringing them to automatism, especially for modelling abnormal and emergency situations It is important for a simulator to have a high-quality mathematical model describing the longitudinal dynamic processes in train and able to assess the lift resistance of each car, since any discrepancy in the actual situation involves inadequate response of a locomotive driver when emergency situations occur in real life [15,21].

Purpose
To ensure the stability the limit values for certain indicators are set.These indicators determine the conditions of rail top wheeling-in.One of such conditions is the ratio of lateral and vertical forces acting on the wheel.However, if the above indicators exceed the permissible limits, it does not mean that derailment will take place; it is the necessary but not sufficient condition for the wheel running off the rails [12].
When designing the new and upgrading the existing freight cars one of the mandatory conditions, which determines their suitability for use, is the fulfilment of conditions for ensuring the car resistance factor against the lift caused by longitudinal forces in the train.
The purpose of the analytical study of the relationship between the longitudinal force acting on the lightweight car, lateral and vertical forces acting in the contact area of the wheel flange and the rail with the lift resistance factor value is to provide simple functional connections between them.

Methodology
Wheel stability against re-railing is determined by two main factors: the ratio of the lateral interaction force to the vertical load on a wheel and by the angle of rail top side edge wheeling-in, which depend on dynamic processes of vehicle movement in the rail track and the longitudinal forces acting on the car in train.
As a result of the use of certain driving modes the train can evoke large longitudinal dynamic forces both tensile and compressive ones.The effect of the action of the longitudinal forces is defined by their direction and car position relative to the neighbouring vehicles in coupling [2,3,7,8].When placing the cars at an angle to each other in the plan, the transverse horizontal components of the longitudinal forces appear; and in case of difference in height of their automatic couplingsvertical components, which create additional loading or off-loading of the respective wheels of the vehicle [3,6,13].This has an effect on the value of the lateral and vertical interaction forces, defining the conditions necessary for rail top wheeling-in [23,24,25].
Since the set of the cars with central automatic couplings of shock-traction action is a multilink hinge-rod circuit, the instability of equilibrium of this mechanism depends on the location and characteristics of the horizontal and vertical components of the longitudinal forces [3].The force, by analogy with Euler static problem, corresponding to hinge-rod circuit buckling moment, is called buckling (Fig. 2).The calculated buckling forces for freight cars occur the least ones when skewing according to the diagram (a).Consequently, the cars coaxially arranged on a straight track section would not be out of alignment under the action of compressive forces lower than the buckling one [2,3].In fact, due to the meandering motion of vehicles, track gaps, spring, axle and centre plate arrangements, and for other reasons the cars almost always have some oblique settings relating to each other [2,3,6,13].
Despite the fact that the theoretical studies found that, when the compressive force value is lower than the buckling value, the car lift should not occur due to the stability loss of the train as a hinged-rod system, the practice shows that the train derailments occur even at lower values of the longitudinal forces.This is particularly evident for the trains with empty cars, which are characterized by intense hunting in the operational speed range.Therefore, apart from the train longitudinal stability as a hinged-rod system, for the speeds above the critical speed of hunting, in the presence of the longitudinal compressive forces one must consider longitudinal and transverse stability of the motion of the train cars [18,19].
It should be noted that the presence of longitudinal forces not only leads to an increased hunting, but also helps to reduce the vertical loads in the wheel-rail system, that is to reduce the forces that prevent rail top wheeling-in, as well as to a greater turn of carriages with increased angle of rail wheeling-in.In such situations, the rail top wheeling-in resistance is seen not as a direct result of the high value of the longitudinal force, but as a result of intense vibrations of the hunting vehicle.This is particularly evident in the area of transition curves, characterized by additional perturbations of impact nature [2,3,[6][7][8].
Let's consider the car motion in a curve, where there are different on both sides of the car cant angles of rigid rods (bodies) to the track plane φ 1 and φ 2 , the angles of coupling rotation with respect to the track axis ψ 1 and ψ 2 , and the car rotation within the gap in the track at an angle ψ.Herewith the car is set in the track under the diagram (a) -«herringbone» and is subjected to the action of longitudinal compressive forces.We introduce a number of assumptions, simplifying the problem [6].Where G 0 -weight of the car body, kN; Р cf -centrifugal force, kN; α -track inclination angle towards the horizontal, rad; N 1 and N 2 -longitudinal forces acting in a car coupler, kN; Z 1 and Z 2 -vertical components of the longitudinal force, kN; 2a -the distance between the centres of the cotter hole of automatic coupling, m; ∆ 1 -axe level difference of automatic couplings in front of the car, m; ∆ 2 -axe level difference of automatic couplings behind the car, m; Y 1 and Y 2 -horizontal lateral components of the longitudinal force, kN; Since the angles φ 1 and φ 2 , ψ 1 and ψ 2 are rather small, then: X 1 and X 2 -horizontal longitudinal components of the longitudinal force, kN.
Assuming that the turning angles of all cars in a curve are equal, we find: In view of the elastic transverse deformation of the bogie spring sets the formulas (7,8,9) take the following form: where 2L c -car coupled length, m; 2ℓ -truckcentre spacing, m; R -curve radius, m; 2δ 0 -total cross-run play of the car body frame relative to the track axis in the run section, m; L -horizontal distance from the car axis to the cotter hole centre of automatic coupling, m; а -coupler casting length (from the axis of coupling to the end of the shank), m; С h -horizontal stiffness of the spring suspension of a bogie, kN/m.The vertical components of support reactions are indicated in Fig. 3 as R1, R2, R3, R4, the horizontal components of the reactions belonging to the same bogie are taken equal to each other (the assumption is made about the equality of the horizontal lateral stiffness of spring group on one bogie) and denoted Н 1 and Н 2 .
Having formed the equations of equilibrium and compatibility of vertical displacements (vertical stiffness of all the spring sets are equal, and the car frame is absolutely rigid), we determine the reactions of the supports: Where 2b -distance between the points of application of the vertical forces acting on the journal box of the wheel pair on the climbing and non-climbing wheel respectively, kN; h -height of the car body centre of gravity over the bolster supporting plane, m; с -vertical distance from the top plane of the central spring set to the coupling axis level, m; а un -unbalanced acceleration, m/s 2 ; g -gravity acceleration, m/s 2 .
Having considered the balance of non-sprig part of the bogie, we can find vertical P 1 , P 2 and the horizontal force Н f acting on the wheel pair: Having considered the balance conditions of a single wheel pair, we get [6,22]: Where h hs -height from the rail level to the top plane of the central spring set, m; G wp -weight of the wheel pair, kN; Н f -frame force acting on the wheel pair, kN; µ -coefficient of flange friction against the rail top fillet.
Solving the system of linear equations we determine the vertical and lateral cross reactions of rail to the climbing wheel: Where -∆ψ = ψ ψ -difference between the swivel angles of couplings against the track axis in front of and behind the car, rad.
If it is necessary to take into account inertia forces in the curve, the vertical ( 21) and the lateral ( 22) reactions of the rail on the climbing wheel are added to the inertia forces caused by unbalanced acceleration: ( ) Using analytical expressions for determining angles (10,11,12), we write the obtained depend-ences for the vertical (21) and the lateral (22) reactions of the rail on the climbing wheel as follows: Where ( ) ψ cur -rotation angle from the curve, rad; .
Substituting the expressions (26) and ( 27) into the formula (24) we can obtain the dependence similar to that described in [6], excluding (31) and taking into account the inertial forces in a curve (32).In addition, we recall dependence for determining the longitudinal forces lift resistance factor excluding (33) and taking into account the inertial forces in a curve (34) which has been studied in [11].
Where P car -car weight, kN; Р in -car inertial force, kN; Р b -vertical load by bogie, which takes into account off-loading under the longitudinal force, kN; N -longitudinal force on automatic coupling during the car compression, kN; 2L ccoupled length, m; 2L -car length over head stock, m; 2ℓ -truck-centre spacing, m; h cp -height of the centre plate working plane above the rail top, m; h hs -height of the spring set top above the rail top, m; h cpl -height of the coupling axis above the rail top, m; h c -height of the car gravity centre above the rail top, m. 2b -distance between the points of application of the vertical forces acting on the journal box of the wheel pair on the climbing and non-climbing wheel respectively, kN; 2S -distance between the taping lines, m; R -curve radius, m; 2δ 0 -total cross-run play of the car body frame relative to the track axis in the run section, m; а -coupler casting length (from the axis of coupling to the end of the shank), m; µ -coefficient of flange friction against the rail top fillet; βangle formed by the straight part of the wheel flange profile contour to the track plane; а un -unbalanced acceleration, m/s 2 ; α and γ -coefficients taking into account the effect of longitudinal buckling forces, leading to a relative imbalance of cars and the coupling axes in the plan.

Findings
To assess the impact of the longitudinal forces value on the lift resistance factor Fig. 6 presents the calculation results for the empty gondola car motion, model No. 12-532, on 250 m radius curve with 150 mm rise and cross starting of car underframe relating to the track axis in 50 mm guiding section [12].The calculations were made in such a curve excluding the inertial forces from outstanding acceleration and taking into account the unbalanced acceleration with the permissible speed of 65 km/h [14,17].
Fig. 6 shows the calculation results for lift resistance factor of the empty car depending on the value of longitudinal forces caused by the neighbouring cars with the difference in coupling heights of 100 mm.The horizontal lines show the allowable value of the lift resistance factor (1.2 [12]) and the value of this factor (1.0), which provides the necessary condition for instability.
The diagrams in Fig. 6 show that the car speed has some significant impact on the lift resistance factor value only at small values of the longitudinal forces, in case of which the lift is impossible.Therefore, as noted previously [3] and as follows from Fig. 6, the impact of the car speed on the longitudinal force lift resistance factor can be neglected.
Fig. 7 shows the dependence of the lift resistance factor obtained using 2 different design diagrams.The dotted line corresponds to the design diagram, which takes into account the loss of stability of the train, as the hinge-rod system [6].The solid line -to calculation diagram, shown in Fig. 3.
This figure proves that, according to the results obtained from the formula (33), the lift resistance is not provided at the force 605 kN and by the formula (31) -even at the force 530 kN.A necessary condition for the loss of stability is fulfilled, in the first case, when the value of the longitudinal force is 800 kN, in the second case -695 kN.Thus the results obtained by the formula (31) impose more stringent requirements to ensure the car lift resistance.
As it is known, the lowest values of the lift resistance factor are peculiar to the cars with small coupled length, truck-centre spacing and unladen weight.In future, the comparative analysis is made only for this category of freight cars [4,5].The calculations were performed according to the dependencies (31) and (33) -excluding the inertial forces by unbalanced acceleration, and according to the dependencies (32) and (34) -taking into account the inertial forces by unbalanced acceleration when driving with permitted speed [17].The results are summarized in Table 1-5.The results of calculations in Tables 1-5 show that for all the above cars the values of the lift resistance factor obtained by the proposed formula (31) are smaller than the standards [12].And for some cars the value К res is lower than the permissible value.
It follows that, even in the absence of «lift» according to the classical train stability theory, as the hinge-rod system, the presence of the longitudinal compressive forces may become a factor provoking cars derailment.Thus the main reason of wheel climbing on rails is the combination of processes of dynamic interaction between the hunting vehicle and the track with simultaneous action of longitudinal compressive forces, and not the destabilization of train as a hinged-rod system.

Originality and practical value
This study provides the technique of determining the lift resistance factor by longitudinal forces, which is somewhat different from the above mentioned one [3,6], as well as evaluates impact of rolling stock speed on this factor.The developed proposals for the refinement of existing methods for determining the resistance factor of the car lift by longitudinal force will increase the stability margin.
The research results have found their scientific use in a number of publications in the special and scientific journals, presentations at scientific meetings.

Conclusions
As a result of analytical research there are obtained the dependencies of longitudinal forced lift resistance factor in the train with and without consideration of the speed of its movement.It is shown that as the value of this factor gets closer to the buckling area the influence of speed decreases.
From these studies one can conclude that because of cars hunting their lift is possible even when the train as hinge-rod system does not lose stability.
Thus, the calculation results allow an objective assessment of the effect of the longitudinal force and the freight car speed on the lift resistance factor value.

Fig. 1 .
Fig. 1.The histogram of traffic accidents caused by locomotive facilities of Ukrainian railways: a − causes of transport accidents; b − transport accidents by type of rolling stock and traction

Fig. 2 .Fig. 3 .Fig. 3 ,
Fig. 2. Design diagram of the train and oblique settings of its elements in the plan under the action of longitudinal compressive forces

Fig. 4 .
Fig. 4. Car layout on a curved track section under the action of longitudinal compressive forces

Fig. 5 .
Fig. 5. Calculation diagram of forces acting on the bogie as a result of longitudinal compression