MODELING OF DEVELOPMENT VERTICAL DEFORMATION OF RAILWAY TRACK

<jats:p>Purpose. State of railway track must meet the conditions of safety, comfort and smooth ride. The presence of irregularities deteriorates the dynamics of interaction of track and rolling stock, causes speed limiting, creates the possibility of movement safety violation. This brings up the question concerning the study of the factors leading to the possibility of track irregularities and the process of their development. The purpose of this paper is to analyse the processes of emergence and development of irregularities in the area of unequal vertical elasticity of railway track using mathematical modelling. Methodology. Railroad under the trains works as the system of elastic bodies, so the emergence and development of irregularities can be represented as the transition from elastic to permanent strain. Irregularity development will affect the dynamics of interaction between track and rolling stock not only at the wheel location directly in the area of irregularity, but also at a certain distance beyond. Therefore, to study the development of irregularities, including those along the track, it is necessary to model the process of wheel load movement along the area. The adopted model consists of a wheel set moving on  inertia-free beam and resting on individual supports. It is described by Lagrange differential equations. The work introduced the hypothesis that the level of permanent strain is distributed in proportion to the dynamic deflection derivative. Findings. Location of vertical longwise irregularity does not necessarily reproduce the location of the problem area. While in operation the vertical irregularity extends not only in depth but also along the track, herewith the increase in length is accompanied by the displacement of local maxima and the emergence of new ones. This leads to the development of so-called «pits» when approaching unequal-elastic areas. Originality. The work provides further development of tasks for track and rolling stock interaction modelling, in particular aimed to take into account the unequal elasticity areas and their influence on the formation of the track irregularities. The paper proposes new approaches to modelling the transition from elastic to permanent strain that allows predicting the development of track irregularity sizes depending on the area characteristics. Practical value. The results obtained by the author can be used to determine the schedule for track equal elasticity renovation works, as well as to analyse the measures aimed at the prevention of irregularities in areas with variable elasticity of railway track.</jats:p>

section_content: During the whole period of operation the railway track must meet the desired conditions, especially the possibility to implement the set speeds. It is the practice to assess the track state by the indices of its geometric position. 

During the track operation, even if it is in full compliance with the standards, various geometric irregularities are gradually emerging and developing. Their elimination and prevention are reputed to be the main task of interim track repairs and its current maintenance [11]. 

The presence of significant irregularities deteriorates the dynamics of interaction of track and rolling stock, prevents from comfort riding and even creates the possibility of movement safety violation. When reaching a certain size the irregularities cause speed limiting [14]. This issue is of particular relevance in terms of modern trends of increasing velocity [7, 15], whereas the track maintenance rules in this case are more demanding. 

Many modern scientific papers are devoted to problems related to the study of track irregularities. They cover the question of their influence on the train dynamic performance [8, 10, 17, 20], as well as the means and methods of their measurement and evaluation [12, 16], and the design of measures to strengthen the rail support layers to avoid the emergence of irregularities [18, 19, 21], etc. 

This brings up the question concerning the study of the factors leading to the possibility of track irregularities and the process of their development. Given that railway track under the trains works as the system of elastic bodies, the emergence and development of irregularities can be represented as the transition from elastic to permanent strain And that is emphasized in many scientific studies, the cause of this is usually the areas of local unequal elasticity. 

Unequal elasticity of rail support can occur in different cases. This may be a result of railway track state violations -presence of track depression, unsuitable fastenings, ballast pollution, etc. [1, 20]. The most prone to the unequal elasticity development areas are those with heavy traffic or leaning on weak soils. Besides the track unequal elasticity can be caused by structural features, such as adjacency to non-ballast bridges [13], presence of crossings [6, 22], etc.
section_name: Introduction

section_content: The purpose of this paper is to analyse the processes of emergence and development of irregularities in the area of unequal vertical elasticity of railway track using mathematical modelling.
section_name: Purpose

section_content: Today there are many different methods of physical and mathematical modelling for interaction of railway track and rolling stock. Depending on the problem to be solved, one can use both relatively simple two-dimensional design models and developed models, which include the systems with dozens of equations. Despite the fact that this refers to the interaction of track and rolling stock, still the problems focused on the rolling stock study, and those focused on the railway track study have fundamental differences. Rolling stock models are, in most cases, the systems of motion (vibrations) of the interconnected solids. Typically for the mathematical description of these models use the rail track more appropriate to describe not move through solids, and because of their deformity. Typically such models are mathematically described by D'Alembert's principle. Railway track operation is more naturally described not as motion of solids, but the strains thereof. Therefore, the railway track is more often mathematically described by the models based directly on elasticity or its numerical representations, such as FEM (finite element method) and others. Thus, using FEM, the authors of the works [9, 10] examined the accumulation process of track vertical strain in the experimental section caused by polycyclic application of force. The essence of modelling process described in these works was as follows. The first calculation resulted in sleeper displacement in a vertical plane due to the applied load. The calculation is repeated for subsequent iterations, but each sleeper displacement acquired in the previous calculation is retained in the form of air gap. 

The calculation results of iterative simulation for track settlement are shown in Fig. 1. So the work [10] presents the conclusion that the increasing number of iterations (tonnage pass) leads to the development of track settlement, but the speed of this process over time decreases. 

Increasing irregularity affects the dynamics of interaction between track and rolling stock. Herewith this effect will be significant not only at the wheel location directly in the area of irregularity, but also at a certain distance beyond (due to the gradual stabilization of vibrations and force redistribution among the bogie wheels). Therefore, to study the development of irregularities (including those along the track) it is necessary to model the process of wheel load movement along the area. 

Given that all factors except vertical dynamics are irrelevant for this problem, we adopted a simplified model consisting of a wheel set moving on inertia-free beam (rail), resting on individual supports (sleepers). The modelling took into account the load transmitted from the wheel set, hard and

The mathematical description of the model consists of a system of Lagrange differential equations of second kind and has no fundamental differences from similar works [2, 4]. 

Separation of permanent strains from total ones is a complex scientific problem, especially for multilayer systems such as railway track. 

Establishing direct linear dependence of the permanent strains values on the total ones violates the adequacy of the model, especially under condi-tions of stress, which is much lower than the level of strength (which is more typical for passenger traffic). In this case, the cause of permanent strains will be not the track deflections, but their longwise unevenness. Therefore the hypothesis was adopted that increase in permanent strains for the next step of iteration ( ( )

), determined by passed tonnage (T ), is distributed along the area ( x ) in proportion to dynamic deflection derivative dyn ( )
section_name: Methodology

section_content: The proposed approach allows exploring the process of irregularity emergence caused by the track unequal elasticity and its subsequent development while in service. 

Let's consider the calculation procedure based on the example with actual numerical output data. We assume the railway area with the place of local unequal elasticity, described by linear change of rail support elasticity modulus from 40 to 30 MPa in the middle of the area over 5 m length [5]. This corresponds to the following sequence performed to supports (sleepers):

{ } 40,. . ., 40, 37, 33, 30, 33, 37, 40,. . ., 40

The first calculation is performed for the track without irregularities. Figure 3 shows the modelling results in the form of the area dynamic longwise deflection caused by passing rolling stock. The deflection in the zone of constant modulus of rail support elasticity corresponds to the analytical calculations by the known formula [2, 3]

where P -wheel vertical force acting on the rail; k -relative rigidity factor. The algorithm (1) allowed determining the permanent strains and transferring them to the model as initial geometric track irregularity for the next iteration. Thus, the process of gradual development of irregularities is modelled -Fig. 4. For visual separation of results the iteration sequence in Fig. 4 is shown with omitted intermediate steps. 1 … 6 -sequence of calculation iterations Similar to the considered numerical example there were performed variant calculations for various input data. The study results allow us to establish certain tendencies. The irregularity outline, which is formed by permanent strains, reproduces neither the outline of dynamic rail deflection nor the outline of initial unequal elasticity. The permanent strains acquire a maximum at the entrance and exit of the dynamic irregularity (first it is unequal elasticity zone, which disturbs the trajectory of the wheel passing over it, and then -its merge with geometric (static) irregularity). While in operation the vertical irregularity extends not only in depth but also along the track, herewith the increase in length is accompanied by the displacement of local maxima and the emergence of new ones. Development of irregularities with a gradual shift of peaks leads to the fact that it itself becomes more significant factor of additional force interaction compared to the original one -local unequal elasticity, causing further expansion and shift of irregularity. The result is that the geometric irregularity does not always strictly coincide with the location of cause of its formation. 

In the above example (Fig. 4 ) the cause of irregularity formation was unequal elasticity zone. But once irregularity reaches a certain size, it is the one that defines the dynamics of interaction between track and rolling stock and, consequently, further development of the process. To demonstrate this observation the iterations in the previous example were stopped at formation of II-degree deflection [14] -the line «3» in Fig. 4. Further calculations include the constant modulus of rail support elasticity along the whole area. The resulted outline of the irregularity is shown in Fig. 5. For comparison, this figure also imposed irregularity from the previous example (line «6» in Fig. 4 ). Figure 5 shows that local irregularity area had already no significant effect on the development of irregularities. Fig. 6. Modelling of track irregularity in the crossing area:

In some cases, the formation of track unequal elasticity zone might arise not from maintenance errors, but from design peculiarities, for example, transition area from ballast track to the bridge [13] or crossing zone [5, 6]. Based on the proposed method the development of vertical irregularities in the crossing zone was modelled. The crossing zone in the output data is a track area with high increase in modulus of rail support elasticity [6] :

{ } 40,. . ., 40,120,. . .,120, 40,. . ., 40 U =. (4)

The above example shows that the presence of the crossing, having formed a zone of increased modulus of rail support elasticity, provokes the emergence of track irregularity. The maximum amplitude of such irregularity will be located at the beginning and at the end of the formation. Over time of area operation, the irregularity will develop, along the track as well, that will lead to the emergence of «gaps» outside the structure. These findings correlate with the results of statistical processing of field measurements of irregularities ahead of non-ballast bridges and in the crossing area, that are given in the works [5, 6, 13] and others.
section_name: Findings

section_content: The work provides further development of tasks for track and rolling stock interaction modelling, in particular aimed to take into account the unequal elasticity areas and their influence on the formation of the track irregularities. 

The paper proposes new approaches to modelling the transition from elastic to permanent strain that allows predicting the development of track irregularity sizes depending on the area characteristics. 

The obtained results can be used to determine the schedule for track equal elasticity renovation works (current maintenance, complex repairs, intermediate overhaul), as well as to analyse the measures aimed at the prevention of irregularities in areas with variable elasticity of railway track.
section_name: Originality and practical value

section_content: One of the main causes of geometric irregularities should be considered track unequal elasticity. 

Location of vertical longwise irregularity does not necessarily reproduce the location of the problem area. 

While in operation the vertical irregularity extends not only in depth but also along the track, herewith the increase in length is accompanied by the displacement of local maxima and the emergence of new ones. 

The irregularity amplitude increase intensity in place of its original formation is reduced over operating time, but the process evolves in other places, resulting in development of so-called «pits» when approaching unequal-elastic areas.
section_name: Conclusions

section_name: LIST OF REFERENCE LINKS

section_content: Гавриленко, А. К.
section_name: 1.

section_content: Цель. Состояние железнодорожного пути должно соответствовать безопасности движения, плавности и комфортабельности движения. Наличие неровностей ухудшает динамику взаимодействия пути и подвижного состава, является причиной ограничения скорости движения, создает возможность нарушения условий безопасности движения. Возникает вопрос исследования факторов, которые приводят к возможности образования неровностей пути, и процесса их развития. Целью данной работы является анализ процессов возникновения и развития неровностей в зоне вертикальной неравножесткости железнодорожного пути с использованием математического моделирования. Методика. Железнодорожный путь под поездами работает как система упругих тел, поэтому появление и развитие неровностей можно представить как процесс перехода от упругих к остаточным деформациям. Увеличение размеров неровности будет влиять на динамику взаимодействия пути и подвижного состава не только во время нахождения колеса непосредственно в зоне неровности, а и на определенном расстоянии за ее пределами. Поэтому для исследования развития неровности, в том числе по длине пути, необходимо моделировать именно процесс движения колесной нагрузки по участку. Принята модель, состоящая из колесной пары, которая движется по безинерционной балке, опирающейся на отдельные опоры. Она описывается системой дифференциальных уравнений Лагранжа. Введе-
section_name: МОДЕЛИРОВАНИЕ ПРОЦЕССА РАЗВИТИЯ ВЕРТИКАЛЬНЫХ ДЕФОРМАЦИЙ ЖЕЛЕЗНОДОРОЖНОГО ПУТИ