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ЗАЛ1ЗНИЧНА КОЛ1Я
UDC 625.1.033.34
D. M. KURHAN1*
1 Dep. «Track and Track Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 373 15 42, e-mail [email protected], ORCID 0000-0002-9448-5269
MODELING OF DEVELOPMENT VERTICAL DEFORMATION OF RAILWAY TRACK
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.
Keywords: track; track irregularity; rail deflection; unequal elasticity of track; interaction of track and rolling stock; railroad crossing
Introduction
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].
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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.
Purpose
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
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
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dissipative connections between bodies of the model, the possibility to set different hardness for each support and geometric outline of the rail in the vertical plane, Fig. 2.
Fig. 1. Calculations results of iterative FEM modelling of track settlement [10]:
P - external load; Y - settlement; X - distance from the force application point; N -iteration number
Fig. 2. Design model of interaction between track and rolling stock:
1 - wheel; 2 - rail; 3 - sleepers; 4 - sleeper support
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 (Azperm (x)), determined by passed
tonnage (T), is distributed along the area (x) in proportion to dynamic deflection derivative
dyn
( x)
Az (x)-
perm v /
dzdyn( x)
dx
Azperm (x) < f (T), 3x, Azperm (x) = f (T)
(1)
Findings
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):
U = {40,..., 40, 37, 33, 30,33, 37, 40,..., 40}. (2)
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]
( ) Pk
z( x) =-,
2U
(3)
where P - wheel vertical force acting on the rail; k - relative rigidity factor.
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E
£
-1,5
с
HM
<U -2
<U
Q
-2,5
45 50 55 60
Area length coordinate, m
«►1—2
Fig. 3. Dynamic rail deflection for irregularity-free areas with local unequal elasticity:
1 - unequal elasticity zone; 2 - rail deflection
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.
E 0
E
.c -5
4M
a
Ol
■о -10
>
HH
(0 -15
3
Ш
IK -20
и
-25
45 50 55 60 65
Area length coordinate, m
70
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.
^►unequal elasticity zone -1 -2 -3 -4 -5 -6
Fig. 4. Modelling of track irregularity development in the area of local unequal elasticity:
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
F 0
E
.с -5
Q.
<U
■o -10
rz -15
3
Ш
a -20
-25
45
70
50 55 60 65
Area length coordinate, m
► unequal elasticity zone —1 -2 —3
Fig. 5. Modelling of track irregularity development for different variants of area state: 1 - initial irregularity; 2 - irregularity formed in view of effect of initial unequal elasticity zone; 3 - irregularity, formed by the output of the conditions of area equal elasticity restoration
-1
5
5
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5 0 -5 -10 -15 -20
55 60 65 70 75 80 Area length coordinate, m
Fig. 6. Modelling of track irregularity
in the crossing area: 1 ... 6- sequence of calculation iterations
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]:
U = {40,..., 40,120,..., 120, 40,..., 40} . (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.
Originality and practical value
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.
Conclusions
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.
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Д. М. КУРГАН1*
Каф. «Колш та колшне господарство», Дншропетровський нацюнальний ушверситет зал1зничного транспорту 1меш академжа В. Лазаряна, вул. Лазаряна, 2, Дншропетровськ, Украша, 49010, тел. +38 (056) 373 15 42, ел. пошта [email protected], ORCID 0000-0002-9448-5269
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету залiзничного транспорту, 2016, № 1 (61)
МОДЕЛЮВАННЯ ПРОЦЕСУ РОЗВИТКУ ВЕРТИКАЛЬНИХ ДЕФОРМАЦ1Й ЗАЛ1ЗНИЧНО1 КОЛП
Мета. Стан залiзничноl коли повинен вiдповiдати умовам безпеки руху, плавностi й комфортабельносп 1зди. Наявшсть нерiвностей погiршуe динам^ взаемодп коли та рухомого складу, стае причиною обмежен-ня швидкосп руху, створюе можливiсть порушення умов безпеки руху. Постае питання дослвдження факто-рiв, що призводять до можливосл утворення нерiвностей коли та процесу 1х розвитку. Метою дано! роботи е аналiз процесiв виникнення та розвитку нерiвностей у зош вертикально1 нерiвнопружностi залiзничноl коли iз застосуванням математичного моделювання. Методика. Залiзнична колiя пiд по1здами працюе як система пружних тiл, тому поява та розвиток нерiвностей можна представити як процес переходу ввд пружних до залишкових деформацiй. Збiльшення розмiрiв нерiвностi буде впливати на динашку взаемоди коли та рухомого складу не тшьки шд час розташування колеса безпосередньо в зонi нерiвностi, а й на певнiй ввдсташ за И межами. Тому для дослвдження розвитку нерiвностi, в тому чи^ по довжинi коли, необхвдно моделювати саме процес руху колюного навантаження по дiлянцi. Прийнята модель, яка складаеться iз колюно1 пари, що рухаеться по безшерцшнш балш та опираеться на окремi опори. Вона описуеться системою диференцшних рiвнянь Лагранжа. Введена гiпотеза, що рiвень залишкових деформацiй розподiляеться пропорцiйно похвднш динамiчного прогину. Результати. Розташування вертикальное' нерiвностi по довжинi не обов'язково повторюе мюце положення проблемно1 дмнки. З часом експлуатаци вертикальна нерiвнiсть поширюеться не тiльки в глибину, а й уздовж коли, причому збiльшення довжини супроводжуеться змiщенням положення локальних максимумiв та появою нових. Це призводить до розвитку так званих «ям» на пiдходi до нерiвнопружноl дiлянки. Наукова новизна. Набули подальшого розвитку задачi моделювання взаемоди коли i рухомого складу, зокрема для врахування дiлянок нерiвнопружностi та 1'х впливу на утворення нерiвностей коли. Запропонованi новi пiдходи щодо моделювання процесу переходу вiд пружних до залишкових деформацш, як1 дають змогу прогнозувати розвиток розмiрiв нерiвностей коли в залежностi ввд характеристик дшянки. Практична значимiсть. Отриманi автором результати можуть бути використанi для визначення термiнiв призначення ремонтних робгг iз оновлення рiвнопружностi коли, а також для аналiзу заходiв, спрямованих на запобiгання розвитку нерiвностей у зонах iз змiнною пружнiстю залiзничноl коли.
Ключовi слова: залiзнична колiя; нерiвнiсть коли; деформащя коли; нерiвнопружнiсть коли; взаемод1я коли i рухомого складу; залiзничний пере1зд
Д. Н. КУРГАН1*
1 Каф. «Путь и путевое хозяйство», Днепропетровский национальный университет железнодорожного транспорта имени академика В. Лазаряна, ул. Лазаряна, 2, Днепропетровск, Украина, 49010, тел./факс +38 (056) 373 15 42, эл. почта [email protected], ОЯСГО 0000-0002-9448-5269
МОДЕЛИРОВАНИЕ ПРОЦЕССА РАЗВИТИЯ
ВЕРТИКАЛЬНЫХ ДЕФОРМАЦИЙ ЖЕЛЕЗНОДОРОЖНОГО ПУТИ
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-
Наука та прогрес транспорту. Вкник Дншропетровського нацюнального ушверситету затзничного транспорту, 2016, № 1 (61)
на гипотеза, что уровень остаточных деформаций распределяется пропорционально производной динамического прогиба. Результаты. Расположение вертикальной неровности по длине не обязательно повторяет место положения проблемного участка. Со временем эксплуатации вертикальная неровность распространяется не только в глубину, но и вдоль пути, причем увеличение длины сопровождается смещением положения локальных максимумов и появлением новых. Это приводит к развитию так называемых «ям» на подходе к неравножесткому участку. Научная новизна. Приобрели дальнейшее развитие задачи моделирования взаимодействия пути и подвижного состава, в частности, для учета участков неравножесткости и их влияния на образование неровностей пути. Практическая значимость. Полученные автором результаты могут быть использованы для определения сроков проведения ремонтных работ по возобновлению равножесткости пути, а также для анализа мероприятий, направленных на предотвращение развития неровностей в зонах с переменной жесткостью железнодорожного пути.
Ключевые слова: железнодорожный путь; неровность пути; деформация пути; неравножесткость пути; взаимодействие пути и подвижного состава; железнодорожный переезд
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Prof. V. D. Petrenko, D. Sc. (Tech.); (Ukraine); Prof. E. I. Danilenko D. Sc. (Tech.) (Ukraine)
recommended this article to be published
Received: Dec. 03, 2015
Accepted: Feb. 04, 2016
