On the Collision of Railcars as an Interaction of Soliton-like and Shock

Wave-like Perturbations

A.T. D’YACHENKO1,2*, M.S. ABU-KHASAN1,

1Emperor Alexander I St. Petersburg State Transport University,

190031, 9 Moscovsky pr., Saint Petersburg,

RUSSIA

2B.P. Konstantinov Petersburg Nuclear Physics Institute of National Research Center “Kurchatov Institute”, 188300,

1 Orlova Roscha str., Gatchina,

RUSSIA

Abstract: —Several new mathematical issues have been set for the development of high-speed transport, which can be solved in

the framework of hydrodynamics to describe the process of hydrodynamics, creating effective rolling stock dampers. which

requires the improvement and development of the corresponding mathematical apparatus. In this work, we use a hydrodynamic

approach to find the density distributions of matter during railcar collisions at high speeds, which is important in light of the

problems of high-speed transport. In our approach, we found an analytical solution to the obtained hydrodynamic equations for

the one-dimensional case. The equations under study were obtained taking into account nonequilibrium processes. To find a

solution to the hydrodynamic equations, the shock wave approximation is used, similar to the soliton solutions we considered

earlier. Taking into account possible deviations from the results of a one-dimensional problem is considered. Such a reduction of

solutions of hydrodynamic equations to shock waves has not been considered previously and may be of interest for a wide variety

of applied problems. The resulting consideration of railcar collisions is important for solving problems of transport safety and

technospheric safety.

Keywords: — High-speed rail transport, hydrodynamics, railcar collision, analytical solution for slab collisions.

Received: March 19, 2024. Revised: September 6, 2024. Accepted: September 29, 2024. Published: November 19, 2024.

1. Introduction

large number of different physical and mathematical

problems of high-speed rail transport ( see, for example,

[1-5]) are solved using the apparatus of the equations of

hydrodynamics.

In [6–8], we reduced the problem of layer-slabs collision to

a description of the interaction of Korteweg–de Vries solitons.

Here we have obtained a description of the propagation of

shock waves [9-11] for disturbances of arbitrary amplitude,

which can be used in the calculations of dampers (see, for

example, [5, 12–15]) and construction equipment [16–20].

These are calculations of hydraulic transmissions, which

have been developed for a long time, but with the increase in

the speed mode of rolling stock and with the consideration of

friction received a new continuation [21]. To create effective

wheel dampers, vibration dampers, and shock absorbers,

hydrodynamics is also used [2-4], which requires improvement

by taking into account the nonlinearity and non-equilibrium of

the damping process at high speeds. Linear equations are used

in [22], and simplified hydrodynamic equations are used in

[23]. This can be developed further and refined in detail for a

nonlinear compressible medium within our hydrodynamic

approach.

The development of the hydrodynamic approach can be

applied to describe nonlinear dynamics in the calculations of

bridges on high-speed electric transport lines [16], and can

also be used in the analysis of the stability of transport

structures in extreme conditions [17].

In our works [6-8], nonequilibrium hydrodynamics was

proposed. And this can be extended to a wide area of technical

applications and used in the design of wheel shock absorbers,

pipes, transport structures, bridges and other objects of

transport and construction engineering (see, for example, [1-

16.23]) in the light of the problems of high-speed transport,

since we proposed a rigorous mathematical approach.

The current stage of development of railway transport in

Russia and the World is characterized by an increase in the

speed of passenger trains while the state of the railway

infrastructure remains unchanged, which leads to increased

risks to the life and health of passengers in the event of

emergency situati0ons. The most dangerous accidents are

longitudinal collisions of passenger trains with obstacles on the

track, which reflect 99.2% of registered cases of emergency

collisions on Russian railways. The adoption of the Strategy

for the Development of Railway Transport until 2030, which

provides for the production and commissioning of high-speed

and high-speed rolling stock, makes the problem of ensuring

the safety of railway passenger transportation increasingly

urgent . In this regard, the task of increasing the safety of

passenger railcars during longitudinal collisions is a priority

DESIGN, CONSTRUCTION, MAINTENANCE

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A. T. Dyachenko, M. S. Abu-Khasan

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direction for the development of new generation railway

rolling stock . The most effective way to improve the safety of

railway transportation is the development and implementation

of mechanical safety systems for passenger railcars, based on

the use of special destructible elements that absorb the energy

of a train colliding with an obstacle Thus, the task of

developing a methodology for determining the parameters of

security systems passenger railcars and their rationale are

relevant [24] .

Consideration of the collision of high-speed railcars is

important for problems of transport safety and technospheric

safety. The purpose of the work is to develop and theoretically

substantiate technical solutions for ensuring mechanical safety

of passenger railcars in case of longitudinal collisions

Next, Section 2 examines the model used to describe the

collision of rail cars within the framework of the

hydrodynamic approach, then in Section 3 a solution to the

proposed equations is found using shock waves and the results

of the collision of railcars are analyzed in order to determine

the degree of impact of the collision of railcars on their

condition. In Section 4, the main conclusions of the work are

presented.

2. The Model

. In the nonequilibrium case the equations of long-range

hydrodynamics [6,7] are obtained, which in the one-

dimensional case have the form for finding the density

ρ( , )xt

and velocity

υ( , )xt

ρ (ρυ) 0,







(1)

( ρυ) ( ρυ ) 0

m m P

  





. (2)

The pressure is [8]:

(/ρ) ρ

(ρ ρ ) α

(1/ ρ)

PK x





     





, (3)

where

is the equilibrium density and

K mcs



is the

compression modulus ,

αρ0

2mcso



(m)2, and the speed of

sound is

cs

3 103 m/s.

Here we consider the propagation of perturbations of

arbitrary amplitude using the equations of hydrodynamics (1)-

(3). Integrating these equations over the density jump,

assuming the speed of shock wave is equal to the speed of

sound and taking into account the expression for pressure (3),

we obtain an equation for the density. Also, as before with

Korteweg–de Vries solitons, we can integrate over the length

of the layer and take into account the propagation of the shock

wave front and its reflection from boundaries. Since for the

maximum, a wave equation is obtained from the equations of

hydrodynamics that admits the d'Alembert solution [8].

As a result of integration we obtain

(ρ ρ )

1 1 1

ρ ρ' ρ 4 λ 1 exp(λ( )) 1 exp(λ( ))

021

L L x l Dt x l Dt





   



     





, (4)

where

ρ1

is formula (5),

λ / αK

and

is the

boundaries of the railcar, and

is its size,

(ρ υ )

200

2ρ/

(ρ ρ )

D K m



. (5)

An approximate solution of one-dimensional

hydrodynamic equations using shock waves can be used in

calculations of dampers for railcars and construction

equipment. In our works, nonequilibrium hydrodynamics was

proposed. And this can be extended to a wide range of

technical applications

3. Results

Modern technologies of computer modeling and numerical

calculation of the stress-strain state of structures under the

influence of excess shock loads make it possible to predict

with acceptable accuracy the consequences of frontal

collisions and judge the impact resistance and damageability

of locomotives. When designing new generation locomotives,

it is expected that computer technologies will be widely

introduced, which, unlike setting up full-scale experiments

(crash tests), turn out to be less costly economically, are able

to take into account important features of the behavior of the

structure and its material under shock loading, and make it

possible to judge the effectiveness of the adopted technical

solutions already in the early stages of design. As a result,

deadlines are reduced and its quality increases. Consequently,

improving methods for modeling and calculating the impact

resistance of locomotives, taking into account large

deformations of the structural material, impact application of

loads and contact interaction of colliding objects, is an urgent

task and is of scientific and practical interest.

Figure 1 shows profiles of the relative density (

ρ / ρ

) for

the interaction of two rod cars in a system of equal speeds,

when the train begins to move off at an initial velocity of

100





m/s at times

1;2;3;4;5;6;7;8;9t

ms.. Figure 2

shows profiles of the relative density (

ρ / ρ0

) for the

interaction of two rod cars in a system of equal speeds, when

the train begins to move off at an initial velocity of

300





m/s at times

1;2;3;4;5;6t

ms.

Figure 3 shows profiles of the relative density (

ρ / ρ0

) for

the interaction of two rod cars in a system of equal speeds,

when the train begins to move off at an initial velocity of

500





m/s at times

1;2;3;4;5t

ms. After the initial

compression and formation of a hot spot, followed by

DESIGN, CONSTRUCTION, MAINTENANCE

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A. T. Dyachenko, M. S. Abu-Khasan

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187

Volume 4, 2024

expansion, at the expansion stage a rarefaction is observed in

the center

Fig. 1. Density profiles of the collision of rod cars at an initial

velocity of

100





m/s at times

1;2;3;4;5;6;7;8;9t

Fig. 2. Density profiles of the collision of rod cars at an initial

velocity of

300





m/s at times

1;2;3;4;5;6t

Fig. 3. Density profiles of the collision of rod cars at an initial

velocity of

500





m/s at times

1;2;3;4;5t

We also can be found a simplified solution to the issue in

the two-dimensional case. The equations are obtained from the

hydrodynamic equations by integrating the hydrodynamic

equations over the transverse coordinate, assuming that the

density

(x, t) does not depend on the coordinate y.

Fig.4. Instantaneous collision profiles of identical slabs (solid

lines) at velocity

500





m/s various points

0;2;4;6;8t

for the two-dimensional case, the dashed lines are density

profiles for one-dimensional layers.

Thus, Figure 4 shows the density profiles for collisions of

identical railcars with a longitudinal dimension of

6L

with velocity

500





m/s at time moments

0;2;4;6;8t

ms.

In this case, the results are indicated by solid lines The dashed

lines correspond to the one-dimensional case. It can be seen

that in the two-dimensional case, the oscillations of

compression and rarefaction are stronger. As for the size, the

region of rarefaction turns out to be of the order of the railcar

length (6 m) and, therefore, in accordance with the estimates of

shock absorber parameters [5], does not result in destruction.

When considering the behavior of railcars over a longer

time interval, the propagation of the shock wave from the head

of the train to its end, the reflection of the wave and its reverse

movement to the head railcar can be traced. The process of

wave movement is accompanied by a gradual decrease in the

amplitudes of accelerations and forces in the intercar

connections, which is due to the dissipation of energy in the

absorbing devices of the coupling devices and the structural

material. Consequently, to assess the maximum load of the

load-bearing elements of the train, it is sufficient to perform

calculation studies only for the initial phase of the collision,

when the largest longitudinal dynamic loads are experienced

by the head railcar and the railcar following it

The developed method for calculating longitudinal

vibrations of a train allows simulate dynamic processes in the

composition, estimate speeds, accelerations its units, internal

forces in inter-car nonlinear connections. Methodology used in

refined calculations of the longitudinal dynamics of the train at

collision with an obstacle. It is possible to take into account

real diagrams deformation of energy absorption devices

measured experimentally during destructive tests. Rigidity

characteristics can be specified inter-car connections for other

types of inter-car connections. Methodology allows you to

analyze the parameters of dynamic processes as part of

destructible energy absorption devices placed between railcars.

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4. Conclusions

Thus, in the present work, the nonequilibrium hydrodynamic

approach has been further developed to describe complex

systems on the example of slab collision. The non-equilibrium

approach to the hydrodynamic equations allows describing the

experimental data better than the equation of state

corresponding to traditional hydrodynamics, assuming the

establishment of local thermodynamic equilibrium. In this

description, the isolation of the hot spot was essential. In this

paper, we show that the introduction of dispersion terms does

not violate this representation. During the expansion stage, a

rarefied region is formed in the center of the system. This

consideration was carried out in one-dimensional case and

may be carried in two-dimensional case too.

The reduction of the hydrodynamic equations to the solution

of two Korteweg-de Vries equations in the form of solitons

makes it possible to find an analytical solution to the issue.

As a result of our assessment of the modeling results, a

refined hydrodynamic computer model of railcar collisions can

be selected for further research. In the third part of the work,

we proposed an approach for selecting the parameters of

energy absorption devices [24]. Thus, on the basis of the

“Universal Mechanism” program complex, as a result of

calculations and numerical experiments, for example, the

development of a energy absorption devices design can be

carried out. Prospects for further development of the topic of

structural protection of railway transport may be associated

with the improvement of test scenarios of emergency situations

and technologies for their modeling.

Acknowledgment

Authors express their gratitude to I.A. Mitropolsky for

useful discussions.

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Contribution of individual authors to the

creation of a scientific article (ghostwriting

policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of funding for research presented in a

scientific article or scientific article itself

This approach can be extended to a wide range of

technical applications and used in the design of

wheel dampers, pipes, transport structures, bridges

and other transport and construction equipment in

the light of high-speed transport problems.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article

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