ABOUT THE EVALUATION OF THE LONGITUDINAL FORCES LEVEL EFFECTING THE TRACK DISPLACEMENT AT TRANSIENT MODES OF TRAIN MOVEMENT

ARL DSRS, Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St. 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 776 72 44, fax +38 (056) 776 72 44, e-mail onildpps@gmail.com, ORCID 0000-0002-6537-7461 Dep. «Theoretical Mechanics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St. 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 776 72 44, fax +38 (056) 776 72 44, e-mail onildpps@gmail.com, ORCID 0000-0002-5128-0095 Dep. «Structural Mechanics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St. 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 776 72 44, fax +38 (056) 776 72 44, e-mail lyudmila.urs@yandex.ua, ORCID 0000-0001-5957-6926 Insurance corporation «Таsо – Garantiya», Transportna St.,3, Odesa, Ukraine, 65039, tel. +38 (048) 267 77 10, fax +38 (048) 264 54 56, e-mail arsonov@list.ru, ORCID 0000-0002-6202-2657


Introduction
As the experience of freight trains operation shows, track displacement occurs when tractionbraking running is applied in order to keep given speed of mode, especially on an excessive gradient and downhill length accordingly [2, 3, 6−7, 11−15].

Purpose
Freight trains safety control requires studying the effect of transient mode of their movement on track displacement.

Methodology
Processes of longitudinal forces occurrence of interaction between the track and rolling stock, caused by transient modes of trains movement, were studied by mathematical modeling of longitudinal vibrations of the train using known methods of numerical integration of nonlinear differential equations describing its motion [1,4,5,9,10].
As a simplified model of the train a chain of bodies (vehicles), interconnected by links (inter-car links) was considered. At this it was assumed that each train vehicle consists of a body (solid) and the wheel sets, connected with the body by friction bearings (inelastic link). The elastic properties of the track and wheel sets were not taking into account. It was thought that during the movement of each train vehicle the vertical plane of its symmetry coincident with the vertical plane of symmetry of the assembled rails and sleepers.
At simulation it was also supposed that in the process of translational motion of the vehicle body wheels make pure rolling along the rail without slipping on it. Such wheel motion was considered as compound, consisting of translational motion with rate V C and acceleration a C of center of body masses ( Fig. 1) and rotary motion about the axis of the wheel set with an angular velocity ω and angular acceleration ε. Then during pure rolling , where r -wheel radius (Fig. 2).
It was supposed that longitudinal force Q acts on each vehicle of the train (Fig. 1), which includes a component of the vehicle gravity on the slope of the track, the efforts in the links between vehicles (in inter car links), resistance force of translational motion, for example, from the wind load.  It can be shown that the dynamic equation which describes the motion of the train vehicle in these cases has the form: where v m -body mass of the vehicle, N -wheel set number of the vehicle, ws O I -inertia axial moment of the wheel set.  (Fig. 2), one can change with corresponding moment of forces pair, one of which is attached to the wheel axle, and the other -to the contact point of a wheel and a rail (Fig. 3).
Then each of these moments can be expressed through the moment of the relevant force couples -it is resistance force to motion from friction in the bearings of the vehicle.
Then acceleration of masses center of the vehicle may be expressed as Interaction forces between a wheel and a rail in cases in question are the friction forces arising in the contact point of a wheel and a rail (Fig. 3); at this, if w fr F -it is friction force, acting on a wheel from the rail side, then r fr F -it is friction force, acting on a rail from the wheel side; because the action is a reaction, then r In order to find out what determines the frictional force ws fr F that acts on wheel set, it is necessary to make a dynamic equation of rotational motion of wheel set about its axis: where ws fr F r ⋅ -it is net moment relatively to the axis of wheel set of the friction forces applied to wheel set from the rails side. Then the expressions for the determination the total frictional force acting on the wheel set and a vehicle have the form: F , which depends on the acceleration of the vehicle. Acceleration values of vehicles can be significant at transient modes of train movement. That is why it seems to be interesting to investigate the processes of longitudinal forces rise of interaction between a track and rolling stock at transient modes of train movement and primarily which are caused by their braking.

Findings
Regenerative (electric braking locomotives) and pneumatic braking of the train with a speed of 40 km/h on horizontal sections of the track and slopes were simulated. In some cases, the train before braking was pre-compacted, in others it was extended.
It was assumed that the train consists of 50 four-homogeneous gondola cars, weight 80 tons and four locomotives, type VL-11. Joint of three As one should expect, the highest level of longitudinal forces and accelerations occur when regenerative braking of prior extended trains in the rear end sections of the train.
The dependences of the dynamic additives from motion time for the 1 st , 4 th , 26 th and 52 nd vehicles during regenerative braking in prior extended and pre-compacted trains correspondingly are shown in Fig. 8−9. The total dynamic additive curve (red line) and the braking force for the entire train are shown in Fig. 10−11.  1500 kN (Fig. 4) and the quasistatic ones -950 kN (Fig. 5).
The maximum level of total value additives for the train takes the value of 55 kN at shock processes ( Fig. 8) and 21 kN -at quasistatic ones (Fig. 9).
From the graphs shown in Fig. 8−11, one can conclude that the maximum value of dynamic additive is registered in that section of the train where the greatest value of the longitudinal acceleration occurs. Therefore, at regenerative braking the greatest value of dynamic addition in a prior extended train 2.5 times more of that value which occurs than for pre-compacted train. At braking of the prior extended train the greatest value of dynamic additive occurs in the rear end section, and at braking of the pre-compacted train occurs in front of the train.
The total values of the dynamic additives and braking forces (Fig. 10−11) in the train do not depend on the initial state of the gaps in the intercar links.
Similar dependences during pneumatic braking by the I st stage with discharging of brake of 0.5 atm are presented in Fig. 12−19.  As can be seen from the graphs shown in Fig. 12−17, oscillograms behavior of longitudinal forces and accelerations essentially depends on the initial state of gaps in the intercar links. At braking of the prior extended train the greatest value of dynamic additive occurs in the rear end section, as the greatest acceleration arises there. At braking of the pre-compacted train the greatest value of dynamic additive occurs in the front of the train, as in this case due to lack of shock loads, acceleration of a locomotive substantially exceeds longitudinal acceleration of other vehicles.
Comparison of the results presented in Fig. 8−9 and 16−17, showed that the greatest value of the dynamic additive of the regenerative braking is almost 2 times higher than similar value, obtained during braking by the 1 st stage of the prior extended train and almost 6 times higher at regenerative braking of the pre-compacted trains. The highest total value of dynamic additive at braking by the 1 th stage occurs in the rear end of the train regardless from the initial state of train set.
When comparing the results in Fig.10−11 and Fig. 18−19 it is clear that the greater value of total dynamic additive to longitudinal forces of interaction between a track and rolling stock for the entire train occurs during regenerative braking, and 2 times higher than similar value arising at pneumatic braking. It is evidence that the regenerative braking is more dangerous for track displacement.
It should be also noted that regardless of the braking type (regenerative or pneumatic) and initial state of gaps in intercar links, dynamic additive value was much less than arising braking forces. That is why the level of longitudinal forces arising in intercar links at the considered modes of movement has little effect on the track displacement.

Originality and practical value
The longitudinal loading of freight trains with regenerative braking and pneumatic one was investigated. The impact of initial state of the train and the different modes of braking on the dynamic additive to the longitudinal forces of interaction between a track and rolling stock was estimated. It may affect the assembled rails and sleepers. Obtained results can be used to select the rationale braking modes of freight trains, especially downhill length, from a position to prevent possible track displacement.

Conclusions
Obtained results show that the dynamic additive to longitudinal forces in the wheel and rail interaction depends on the occurring accelerations.
The total value of the dynamic additive was greater at the regenerative braking and does not depend on the initial state of a train set. The level of total dynamic additive in the train was much less than the level of resulting braking forces. Therefore, the level of longitudinal forces in intercar links has little effect on the track displacement.