MATHEMATICAL MODEL OF VIBRATIONS OFA LOCOMOTIVEBOGIE FRAME STRUCTURE OF COMPLEX CONFIGURATION AT INCREASEDMOTION SPEED IN TRANSPORT ENGINEERING Текст научной статьи по специальности «Строительство и архитектура»

155М 2181-953Х

ЖЕЛЕЗНОДОРОЖНЫЙ ТРАНСПОРТ:

АКТУАЛЬНЫЕ ВОПРОСЫ И ИННОВАЦИИ

№ 1-2 2019

Железнодорожный транспорт: актуальные задачи и инновации

Издается с 2019 года

Редакционный совет:

Хосилов Х.Н., доц. Расулов М.Х., проф. Адилходжаев А.И., проф. Блажко Л.С., проф. Бочков К.А., академик Юсупбеков Н.Р.

Редакционная коллегия:

Главный редактор - проф. Адилходжаев А.И.

Заместитель главного редактора - проф. Амиров С.Ф.

Ученый секретарь - доц. Каримова Ф.Ф.

Члены редакционной коллегии:

Академик Аллаев К.Р., проф. Акимаса Фудживара (Япония), проф. Ишанходжаев А.А., проф. Кондращенко В.И. (Россия), проф. Куанышев Б.М. (Казахстан), проф.

Мансуров Ю.Н., проф. Мирахмедов М.М., проф. Никитин А.Б. (Россия), проф.

Петрова Т.М. (Россия), проф. Рахмангулов А.Н. (Россия), проф. Сладковский А.В.

(Польша), проф. Титова Т.С. (Россия).

ТАШКЕНТСКИЙ ИНСТИТУТ ИНЖЕНЕРОВ ЖЕЛЕЗНОДОРОЖНОГО

ТРАНСПОРТА

Учредитель научно-технического журнала «Железнодорожный транспорт: актуальные задачи и инновации» - Ташкентский институт инженеров железнодорожного транспорта (100167, Республика Узбекистан, г. Ташкент, ул. Адилходжаева, дом №1, ком.: 333, тел.+998712990026; e-mail: nauka@tashiit.uz).

В журнале «Железнодорожный транспорт: актуальные задачи и инновации» публикуются наиболее значимые результаты научных и прикладных исследований, выполненных в ВУЗах железнодорожного профиля, других высших учебных заведениях, научно - исследовательских институтах и центрах Республики Узбекистан и зарубежных стран.

Журнал издается 4 раза в год и содержит публикации материалов по следующим основным направлениям:

• Путь и путевое хозяйство;

• Подвижной состав и тяга поездов;

• Электроснабжение, электроподвижной состав, автоматика и телемеханика;

• Организация перевозочного процесса и транспортная логистика;

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• Информационные технологи и информационная безопасность;

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Свидетельство о регистрации средства массовой информации № 0952 выдан Агентством по печати и информации Республики Узбекистан.

Учредитель - Ташкентский институт инженеров железнодорожного транспорта

100165, Республика Узбекистан, г.Ташкент, ул.Адилходжаева д.1.

Тел.: +998 71 299 00 26 E-mail: nauka@tashiit.uz

Railway transport: Actual Tasks and

innovations

Published since 2019

Editorial Council:

Hosilov Kh.N., Assoc. Rasulov M.Kh., prof. Adilkhodjaev A.I., prof. Blazhko LS, prof. Bochkov K.A., Academician N.R. Yusupbekov

Editorial team: Chief Editor - prof. Adilkhodjaev A.I. Deputy chief editor - prof. Amirov S.F. Scientific Secretary - Assoc. Karimov F.F.

Members of the editorial team:

Academician Allaev K.R., prof. Akimas Fujiwara (Japan), prof. Ishanhodjaev A.A., prof. Kondrashenko V.I. (Russia), prof. Kuanyshev B.M. (Kazakhstan), prof. Mansurov Yu.N., prof. Mirahmedov M.M., prof. Nikitin A.B. (Russia), prof. Petrova T.M. (Russia), prof. Rakhmangulov A.N. (Russia), prof. Sladkovsky A.V. (Poland),

prof. Titova T.S. (Russia).

TASHKENT INSTITUTE OF RAILWAY TRANSPORT ENGINEERS

The founder of the scientific and technical journal "Railway Transport: Actual Tasks and Innovations" - Tashkent Institute of Railway Engineers (100167, the Republic of Uzbekistan, Tashkent, Adilkhodzhaev str., Building 1, room .: 333, tel. + 998712990026; e- mail: nauka@tashiit.uz).

In the journal "Railway transport: Actual tasks and innovation" are published the most significant results of scientific and applied research carried out in universities of railway profile, other institutions of higher education, scientific - research institutes and centers of the Republic of Uzbekistan and foreign countries.

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Mass Media Registration Certificate No. 0952 issued by the Press and Information Agency of the Republic of Uzbekistan.

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СОДЕРЖАНИЕ

ПОДВИЖНОЙ СОСТАВ И ТЯГА ПОЕЗДОВ

Рахимов Р.В., Рузметов Я.О. Оценка воздействия подвижного состава с увеличенными осевыми нагрузками на путь и установление условий их обращения на железных дорогах

Республики Узбекистан..................................................................................... 5

Хромова Г., Раджибаев Д. Хромов С. Математическая модель колебаний рамной конструкции тележек локомотивов сложной конфигурации при повышенных скоростях движения для транспортного машиностроения........................................................ 14

ОРГАНИЗАЦИЯ ПЕРЕВОЗОЧНОГО ПРОЦЕССА И ТРАНСПОРТНАЯ ЛОГИСТИКА

Потылкин Е.Н. Выбор режимов взаимодействия железнодорожного транспорта общего

и необщего пользования................................................................................... 28

Кобулов Ж.Р., Баратов Ж.С. Совершенствование технологии обслуживания клиентов

на железнодорожном транспорте......................................................................... 41

Илесалиев Д.И. Методика определения запасов зерна и ёмкости линейного элеватора...................................................................................................... 47

ИНФОРМАЦИОННЫЕ ТЕХНОЛОГИ И ИНФОРМАЦИОННАЯ БЕЗОПАСНОСТЬ

Сиддиков И.Х., Умурзакова Д.М. Синтез адаптивной нейро-нечеткой системы управления нелинейных динамических объектов.................................................... 54

ТЕХНОСФЕРНАЯ БЕЗОПАСНОСТЬ

Халилова Р.Х. Экологическая безопасность дорожно-транспортной системы............... 74

CONTENTS

ROLLING STOCK AND TRACTION OF TRAINS

Rahimov R.V., Ruzmetov Ya.O. Assessment of the impact of the rolling stock with increased axial loads on a way and setting the conditions of their circulation on the railways of the

Republic of Uzbekistan....................................................................................... 5

Khromova G., Radjibaev D., Khromov S. Mathematical model of vibrations of a locomotive bogie frame structure of complex configuration at increase edmotion speed in transport engineering.................................................................................................... 14

ORGANIZATION OF TRANSPORTATION PROCESS AND TRANSPORT LOGISTICS

Potylkin Y.N. The choice of modes of interaction of public and non-public rail

transport....................................................................................................... 28

Qobulov J.R., Barotov J.S. Improvement of customer service technology in

reilway transport............................................................................................. 41

Ilesaliev D.I. Methodology for determination of grain reserves and capacities of a linear elevator......................................................................................................... 47

INFORMATION TECHNOLOGIES AND INFORMATION SECURITY Siddikov I.X., Umurzakova D.M. Synthesis of adaptive neuro-fuzzy control system of nonlinear dynamic objects.................................................................................. 54

TECHNOSPHERE SAFETY

Khalilova R.Kh. Ecological safety of the road transport systems ................................... 74

MATHEMATICAL MODEL OF VIBRATIONS OFA LOCOMOTIVEBOGIE FRAME STRUCTURE OF COMPLEX CONFIGURATION AT INCREASEDMOTION SPEED IN

TRANSPORT ENGINEERING

Khromova G., Radjibaev D., Khromov S.

Tashkent Institute of railway engineering, Tashkent, Uzbekistan

Abstract: A numerical-analytical method for dynamic strength calculation of bogie frames of the locomotive of variable mass and rigidity is presented in the paper; it includes the analysis of the systems of partial differential equations describing the vibrations of the sections of elastic curvilinear elements of variable flexural rigidity with spatial arrangement of sections.These equationsare solvedon a computer by theLaplace methods of operational calculus and further use of iteration methods (the method of the piecewise linear approximations) based on the Fourier and Bubnov-Galerkin methods; numerical studies have been carried out in the MathCAD 14 programming environment.

Key words: bogie frame of the locomotive, dynamic strength analysis.

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ КОЛЕБАНИЙ РАМНОЙ КОНСТРУКЦИИ ТЕЛЕЖЕК ЛОКОМОТИВОВ СЛОЖНОЙ КОНФИГУРАЦИИ ПРИ ПОВЫШЕННЫХ СКОРОСТЯХ ДВИЖЕНИЯ ДЛЯ ТРАНСПОРТНОГО

МАШИНОСТРОЕНИЯ

Аннотация:

Ключевые слова:

В статье представлен численно-аналитический метод динамического расчета на прочность рам тележек локомотивов с переменной массой и жесткостью, включающий в себя анализ систем уравнений в частных производных, описывающих колебания сечений упругих криволинейных элементов переменной изгибной жесткости с пространственным расположением сечений, с учетом повышенных скоростей движения, решение которых осуществляется методами операционного исчисления Лапласа и дальнейшего использования итерационных методов (метод кусочно-линейной аппроксимации) на ЭВМ на базе методов Фурье и Бубнова - Галеркина, численные исследования проведены в среде программирования MathCad 14. рама тележки локомотива, динамический расчет на прочность.

INTRODUCTION

In modern foreign patent and scientific-technical literature, the problems of increasing the reliability and durability of bogie frames and main body frames of locomotives are widely investigated in the process of their design, operation and modernization. The authorsof this paper in [1,2]have investigated the use of various analytical-numerical methods to calculate the dynamic strength of the bogie frames

and main body frames of various vehicles. The frames are modeled as the systems of elastic rods under different static and dynamic loading modes with constant mass and rigidity.

The bogie frame of the locomotive is a complex spatial structure that sustains both static and dynamic loads acting on the bogie and the body units. The greatest forces are applied to the middle part of longitudinal beams, so the sidewall of the bogie frame has a larger cross section in the middle part and a smaller one at the edges. In this regard, we have proposed an analytical-numerical method of design of dynamic strength of the supporting frame of a locomotive bogie under the assumption of the variability of its mass and flexural rigidity along the length under harmonic load.

METHODS OF CALCULATION

The bogie frame of the locomotive is a complex spatial structure that sustains both static and dynamic loads acting on the bogie and the body units. The greatest forces are applied to the middle part of longitudinal beams, so the sidewall of the bogie frame has a larger cross section in the middle part and a smaller one at the edges.In calculation, the bogie frame of the locomotive should be designed for possible unfavorable combinations of the following loads [1,2,3]:

1. staticload;

2. vertical dynamic load;

3. centrifugal forces at locomotive motion along the curve and the wind pressure on the side surface of the locomotive, including the friction force between wheels andrails in the curve;

4. the forces arising from the operation of traction engines;

5. the longitudinal forces arising from the impact on the coupler of an electric locomotive;

6. brakingforces;

7. the forces arising at wheelset rolling out or at locomotive bogie raising when it derails.

Static load corresponds to the axle pressure on the rails or to the load acting on the rails from the impact of the wheelset - 225.4 kN (23 tons) [3].

Vertical dynamic load is considered proportional to static load, the coefficient of vertical dynamics is 1.5 at the estimated speed of motion of 100 km/h [3].

The values of centrifugal forces are taken as high as possible and correspond to the motion of an electric locomotive at a speed of 100 km/h on a curve with a radius of 600 mat asuperelevation of 125 mm. In calculations in theMathCAD 14programmingenvironment, theradiusofthecurverangedfrom125 mto600 m.

The intensity of the wind pressure on the side surface of the locomotive is taken equal to 490 kN.

The magnitudes of the forces acting on the bogie frame at traction engines operation correspond to a maximum coefficient of traction between wheels and rails, equal to 0.35.

Longitudinal impact force on the coupler of electric locomotive is taken equal to 2.5 MPa and is distributed between the elements of the locomotive in proportion to their weight. Due to the balance of spring suspension of the electric locomotive bogie in longitudinal direction, the frames of the bogies are not subjected to skew-symmetric loads when the track is rough.

Calculation of the bogie frame of the locomotive for dynamic strength is conducted in the following order:

1. the magnitudes of the forces acting on the frame are determined and the basic design patterns of loading are formed, these values being the basis for the further calculation;

2. the main design schemes of loading are divided into simpler auxiliary ones: vertical and horizontal, symmetric and antisymmetric loads are considered separately;

3. calculation of auxiliary and main loading schemes for the bogie frame is conducted;

4. for the main loading schemes of the bogie frame, the values of normal stresses are determined. The magnitudes of tangential stresses are not determined in

calculation, due to their small size. It should also be noted that the places (fibers) of the maximum tangential and maximum normal stresses do not coincide.

5. the summation of the stresses in the calculated cross-sections is done at the most unfavorable and possible combination of loads:

5.1. traction on the straight line;

5.2. braking on a straight line.

The stress state of the bogie frame is characterized by the values of the maximum total stresses from the above operating modes of the locomotive, and by the magnitudes of the stresses acting atwheelsets rolling out (when the locomotive is lifted by the bogies).

The authors have carried out a comparative analysis of the stress-strain state of the bogie frame (with and without reinforcing plates)by calculating the fatigue strength coefficients, based on which they can predict the possible period of further useful operation of mechanical part of the locomotive (the VL-80s electric locomotive).

Figure 1,a shows a cross-section of the bogie frame of an electric locomotive VL-80s with calculated cross-sections. Figure 1,b shows the bogie frame fan electric locomotive VL-80 swith calculated cross-sectionsin plan.

The calculated cross-sections of the bogie frame of electric locomotive to determine the stresses are the places of joint of transverse bars to the sidewalls, middle sections of the bars and axle boxes openings. Quasistatic strength analysis is performed by the force method for each design mode. Calculation of geometric parameters for the calculated sections is given in Table 1.

The spatial system of forces acting on the bogie frame of the VL80s electric locomotive under design driving conditions causes the sidewall to bend in two planes, therefore, for the calculated sections, the bending resistance moments alongy and z axes, presented in Table 1, are determined.

Change of the maximum moments of inertia^abyand^crfyalong the beams of the bogie frame of an electric locomotive VL-80s

Fig.1a,b.- The bogie frame of the locomotive VL-80s with calculated cross sections (a) - cross cut, b) in plan: 1 - end barframe rail)of the frame; 2 - the side of the bogie frame (longitudinal beam); 3 - center pin(cross center pin of the bogie frame); 4 - box of ball joint of the pivot; 5 - bracket for hydraulic oscillation damper of axle box spring suspension; 6-large axle box bracket; 7 - bracket for mounting the suspension of the brake lever transmission; 8 - small axle box bracket; 9 - reinforcing platemade of steel sheet 3=12 mm (steel ST3); 10 - reinforcing plate made of steel sheet 3=12 mm (steel

ST3) on the center pin (cross center pin) of the bogie frame of the VL-80s electric locomotive; 11 -reinforcing plate under the roller of the anti-discharging unit; 12 - bracket for a cradle suspension;13 -

bracket for brake suspension.

Analyzing the moments of resistance over the cross-sections of the beams of the bogie frame of an electric locomotive VL-80s W Y andWdy(see Table 1) we can draw the following generalizing conclusion: the most loaded are the sections B-B, D-D, E-E. So, after operating for 20 years or more (the present service life of VL-

80s electric locomotives in "Uzbekistan Temir Yollari" JSC), these sections will not be able to meet the requirements of design modes for dynamic fatigue strength. These sections must be strengthened (modernized) by installing reinforcing figured plates from steel sheets ofST3 grade and by angle bars.

Table 1

Sections on the bogie frame Wabx , cm3 WabY , cm3 Wcdx , cm3 WcdY , cm3

A -A 8084,69 1532 4967,2486 1330

B - B 1221,073 1309,348 1221,073 1309,348

C - C 2760,2326 3079,6198 2938,3122 2529,6877

D - D 2987,102 1517,362 3241,926 1517.362

E - E 712,04 827,552 712,04 827,552

G - G 2665,286 3079,6198 2850,008 2529,6877

For calculation, the parameters ofST3 have been taken, which, according to [5], has the following characteristics: - endurance limit at symmetric cycle a.1= 19.5 MPa;

- yield strength no less than <jt = 240 MPa;

- untimate strengthen = 425 MPa.

Permissible stress on tension (compression): [<] = 115+195 MPa.

The highest value of permissible stresses for the most severe modes of static and shock loading of 2.5 MPa (250 m) at the automatic coupling, in accordance with the requirements of the Central Research Institute of the Ministry of Railways of Russia, is taken to be [8]:

[<] / noy (1)

noy is the standard safety factor;

aT is the yield strength, for ST3of the 1st discharge ofthickness aT no less than 240 MPa.

The smallest permissible value of the actual safety factor in these modes is determined by the formula:

=-К-> 1.1 (2)

^ Pk К )

Here (oy+acm) isthemaximumtotalstressundershock (orstatic)loadof2.5 MPa (250 m)on theautomaticcouplingaxis, takingintoaccounttheweightstresses; fiKis thestressconcentrationfactor.

The calculated bogie frame of the VL-80s electric locomotive has constructive and technological measures to ensure the absence of stress concentrations at the points of transfer of the non-reinforced profile to the reinforced one and at the joint points to the sidewalls of the transverse elements of the bogie frame. This is confirmed by the results of tests of the bogies with sidewalls of shaped steel (two channel bars), where even under vibration load the difference in fatigue limit values for the straight smooth side of the sidewall and for the side with adjacent axle box brackets has not been revealed [8].

These results have been repeatedly tested by experiments, using the low-base strain gauges, the base of which (10 mm) is many times (in some cases - dozens of times) less than the defining dimensions of transition elements.

For the bogie frame units, the design of which is changed simultaneously with the change (lightening) of the sidewall, the coefficient = 1 can be taken. So,

[o] = [ay + oCT ] = 218 Mna (3)

The equivalent supporting framework of the sidewall of the bogie frame of the locomotive is modeled by an elastic rod of variable cross section with a variable mass and flexural rigidity. The difference of the proposed model from the existing ones is an account of the variability of section, mass and flexural rigidity along the length of the equivalent beam; this corresponds to the actual operating conditions. The bogie frame of the locomotive VL-80s, operated in Uzbekistan, is takenfor a specific numerical calculation.

For the model, proposed by the authors, the parameters of the equivalent supporting framework of the sidewall of the bogie frame of an electric locomotive are taken as variable functions (polynomials ofn-th power):

- the linear mass of the sidewall of the bogie frame of the locomotive (kg/m)

is:

mr (X) = mo (ao + a 1X + a 2 X2+...+an X1) (4)

the length of the sidewall of the bogie frame of the electric locomotive is 4,658 mm, and the X coordinate varies within 0 < X < 4,658 m (Figure 1);

- the reduced moment of inertia of the sections of the sidewall of the bogie frame along the axis

X - IX (cm4)is:

Ix (X) = Io (bo + biX + b2X2+...+bnXl) (5)

- the reduced flexural rigidity is:

^h (X) = EIo (bO + b1X + b2X2+...+bXl) (6)

The optimal value of n (polynomial power) is selected using a computer by the piecewise linear approximation method based on the actual dimensions of the sidewalls of the bogie frames of electric locomotives. For the VL-80s electric locomotive, the value of n = 8 (the error ô = 0.001) has been accepted.

To analyze the stress-strain state of the equivalent framework of the sidewall of the bogie frame of an electric locomotive, the differential equations of bending vibrations of straight rods of variable cross section are used[1,3] (assuming the longitudinal and torsional vibrations are small compared to the other components).

(v^d2w{x,t) „ ,v.dW(x,t) a2Ix(x) a2w{X,t)

mT {X)-+ EI X ( X)-+ E-^^--^TL-L= fn\

Ty ! at2 XK ' ax4 ax2 ax2 (7)

= lH ( X, t ) + PM ( X, t )

Where W(X,t)are thebending vibrations of thesections of the sidewall of the bogie frame; P^(X,t)is the dynamic force arising in vertical direction under the effect of various necessary equipment mounted on the bogie frame and the body frame of electric locomotive; for example, from vibrations of the traction electric engine. Dependingon the scheme of its location (a particular type of electric locomotive), the spectrum of a given load along the length X is different;

( X, t) is the function of change in the trackroughness, taken as

T!H(X, t) = { 2C ^L(1 - cosvt) + 2 p-r¡H } • tJh(X) (8)

7ГХ

where ттН(Х)= s i n— ; ю is thefrequency of change in roughness over time;

So,

7ГХ

Пн ( X, t )=s i n— ( С ■ ] 0( 1-е О SO t)+] 0 / OS i ПО t ) (9)

Н

Next, we perform the solutions on a computer using the piecewise linear approximation method and the iteration method. The entire beam (the sidewall of the bogie frame of the VL-80s electric locomotive) is divided into 20 points, at Xchange in the range o<x < 4,658 m; for each of these sections we introduce the coefficients RiK, which are CONST (constant values),^ = 1.2 ... 20, n = 8

Rk = E■ 10-(2b2 + 3b3 +... + 8b8 ■ X8) ; (10)

R2K = 1o-(bo + b X + b2X2+^ + b8 X8) ;

R3K = m (X) = m0 (a0 + aX + a2X2 +... + asX8).

As a result, for a check section equation of the form (7) are obtained with fixed (constant) coefficients:

^++-пн(x,,)+рдк(Хt) (Id

The solution of homogeneous differential equation (11) is performed on a computer by linearization using the piecewise linear approximation method; then the Fourier method is applied to equation (11) with constant coefficients with further application of the Laplace transform in time; numerical studies are performed in the Mathcad 14 programming environment similarly to the methods given in [1,3].

Let's findthesolutionofhomogeneousequation (11) intheform

Rv^ + Rn^ + R 3*^ = 0 (12)

according to the Fourir method[6] in the form:

WK СX,t) = WK (X)-TK (t) (13)

where are the eigenfunctions of the system; bending (transverse) vibrations of the system are obtained in the form [6, p.266; 7, p. 373]:

W k (X) = A k • s h ù) KX + В k • с h ù) КХ + Ck • s in ù bX + Dk • с о s ù b X(14)

where ù K =

N

ьк2

+ùb = 1 bJf+Щ^К (14)

The equation for the eigenfunction (14) and its derivatives are introduced into boundary conditions. To find the coefficients Ak, Bk , Ck , Dk, the system is obtained in the form:

BkcoK2 - Dkù)B2 = 0 Akù)K3 -Ckù)B3 = 0 ElÇA^^shù)^ + Bkù)K2chù)Kl1 — Ckù)B2sina)Bl1 — Dkù)B2cosù)Bl1) = MA ElÇA^^chù)^ + Bk(ûK3 shcd^ — Ck(ûB3 cosù)^ + Dkù)B3sina)Bl1) + +CA [A ks hù)K l± + Bkchù)K l± + Cks in ù)Bl1 + Dkco s ù)Bl1] = 0 (15)

For system of equations (15) to have a solution, its determinant must be zero. From the boundary conditions for the bogie frame of the locomotive o < i < 4,658 m we get

A=

а-21а22а23а2А a31a32a33a34 ^41 ^42 ^43 ^-44

where a11 = 0 ; a12 = ùK2;a13 = — ùB2; a14 = 0;

a21 = ùK3 ; a22 = 0; a23 = — ùB3; a24 = 0; аз 1 = ù)K2shù)Kh; a32 = o>k2cho>Kh; a3 3 = — ù)b2 s inw Bli; a3 4 = — ù) b2 с о s ù)b11

a41 = Elco^chù)^ + Cj^shù)^) a42 = Elco^shù)^ + Сдсксо^; a4 3 = — El ù) B3 с о s ù)Bl1 + CAs ina)Bl1 ; d44 = El ù B3s in ù Bl 1 + CAc о s ù Bl 1 ;

bl = 0; b2 = 0; b3=^;b4 = 0

(16)

To find the coefficients Ak , Bk , Ck , Dkthe Gauss method (using the Cramer rule) is used:

Ak Bk= Ck = ^;Dk =

Д Д Д Д

(17)

Дл =

Ь± % 2a13a14 Ь2а22а23а24 Ь3&32а33а34 Ь4а 42a43a44

AB=

Ar=

"n a12bt "•14 ^21^22^2^24 a31a32b3^34 а41а42Ь4а44

^21^2^23^24 a31b3CL 33CI34

a4lZ?4a43a44 "11

a21a22a23^2

a31a32a33b3 «41^42^43^4

The frequency equation with boundary conditions has the form:

E x ■ s ho)Kl + E 2 ■ cho)Kl + E 3 ■ s ino)Bl + E4 ■ co s o)Bl = 0 at/ = 1 (19) Introduce the notations into equation (20):

Ля =

(18)

с

к - са

К - £ ■I

С

к - ■С

Е ■ I

Е,

К1 ■ (l + со2К / С)+ 3

к

2

с2 + ^ + К2 ■(l + Ç)+Ç ;

с

с

Е

2 =

к1 ■(l+сК/ с )-

+ сК / «Î2+ с2+ к

с

2

Я

E3 = < • de - < - ^2 ;E4 = • coi/os1 - < - < From the orthogonality condition of eigenfunctions, the dynamic displacements TK ( t) are found by equation (24):

(20)

(21) (22)

d2Tk(t) dt2

- X2kTk{t) = CLk(t)

(23)

where

afc(0 =

J n

^Н^МО _ ^2 j Wk(x)dx -mk

= S,

1 /(sm~(C-ri0(l-cosù)t)+ri0Bù)sinù)t)

о —-~-^ ) wkWdx

mT

(24)

where W* (X) arecalculated on a computer for each of the twenty sections using the Gauss method.

By the piecewise linear approximation method in the MathCAD14 program for each specific section I = /T(the bogie frame of the locomotive is divided into 20

points along the length 0 < x < 4,658 m),

^ /\sm~(C-ri0(l-cosoL>t)+ri0l3oL>sinoL>t] \

functionak ( t) = f I-u-—--d\ \Wk (X) dxis constant.

For each individual section in the MathCAD 14 program on the basis of the piecewise linear approximation method, the boundary conditions (conjugation conditions) are taken as equal to [6,7]:

Wk r ig t = Wk l e f t;

dWk _ dWk . md2Wk _ EI d2Wk

dX right dX ie f t dX2 right dX2 le f t

General solution has the form

W (x,t)=Z k=iWk (x)-Tk (t) (26),

where Wk (X)are theeigenfunctions calculatedon a computer for each of the twenty sections using the Gauss method with formulas (14) - (23); dynamic displacements are calculated with equation (24) by integrating along the length using theSimson procedure in the MathCAD 14 programming environment and present a dynamic deflection of the bogie frame of the locomotive over time

CONCLUSION

A mathematical model was developed for frame structures of locomotives of a complex configuration for transport engineering, taking into account the effect of contact dynamic loads;a substantiation of the design scheme of the bogie frame of the locomotive and of design modes available in operation was performed; the basic principles of calculating the maximum stresses of the bogie frame of the locomotive were presented, geometric characteristics of the sections of the bogie frame of the VL-80s electric locomotive were calculated.

As a result, a combined numerical-analytical method was developed based on the iterationmethodand the piecewise linear approximation method. This appliedengineering method makes it possible to perform dynamic calculations of the stress-strain state of the sidewall of the bogie frame of an electric locomotive under harmonic load when it moves along a path with periodic rail roughness. The method can be widely used in design of new and modernization of existing electric

locomotives, taking into account the process of rational computerization.

The tools created and the methods proposed for calculating the modernization parameters can be widely used in design of new structures of electric and diesel locomotives, in planned overhaul and restoring maintenance of operated ones in view of their modernization. This engineering method of calculation allows one to carry out quasistatic and dynamic calculations of the stress-strain state of the bogie frames of the locomotive with modernized reinforced supporting framework. The results of this study will be sent to the Locomotive Operation Management of the JSC "Uzbekistan Temir Yollari "for practical implementation.

References

1. Khromova G.A., Mukhamedova Z.G., Yutkina I.S. Optimization of dynamic characteristics of emergency rail service cars. Monograph. ISBN 978-9943-975-966. - Tashkent: "Fan va tekhnologiya", 2016. - 253 p.

2. Khromova G.A., Kulakhmetov A.P. The study of oscillations of electric locomotive VL-80s at its motion along curved sections of the rail track. // Journal "Vestnik of TashIIT", 2009, No. 6, Pp. 28 - 32.

3. Khromova G.A., Rajibaev D.O., Tuychieva M.N. A numerical-analytical method for dynamic strength analysis of the bogie frames of the locomotive with variable mass and flexural rigidity. / In Proc. of the YI International Scientific and Technical Conference "Locomotives. Electric transport. XXI Century ", volume 1, SPGUPS, St. Petersburg, November 2018, Pp.145-148.

4. Khromova G.A., Khromov S.A., Radjibaev D.O. Estimated substantiationof operation modes of the stand for dynamic testing of the bogie frames of the electric locomotive. / In Proc. of the YI International Scientific and Technical Conference "Locomotives. Electric transport. XXI Century", volume 1, SPGUPS, St. Petersburg, November 2018, Pp. 142-145.

5. Pisarenko G.S., Yakovlev A.P., Matveev V.V. Handbook on strength of materials. 2nd ed., - Kiev: NaukovaDumka, 1988.- 736 p.

6. Babakov I.M. Theory of oscillations: Tutorial / I.M. Babakov .- 4th ed., - M.: Drofa, 2004.-591 p .- (Classical authors of Russian science).

7. Timoshenko S.P. , Young, D.H., Weaver, W. / Trans. from Eng. by L.G. Korneychuk; Ed. by E.I. Grigolyuk. Oscillations in engineering. - M.: Mashinostroenie, 1985.-472 p.

8. Dankovtsev V.G., Kiselev V.I., Chetvergov V.A. Maintenance and repair of locomotives. / Ed. by V.A. Chetvergov, V.I. Kiselev. M: GOU "Teaching-Methodical Center for Education in Railway Transport", 2007.-558 p.

Сведения об авторах / Information about authors

Galina KHROMOVA, Davran RADJIBAEV*, Sergey KHROMOV

Tashkent Institute of Railways Engineering Adylkhodjaev 1, 100167, Tashkent, Uzbekistan * Corresponding author. E-mail: davronmail@bk.ru

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Темир йул транспорти: долзарб масалалар ва инновациялар

Илмий-техник журнал

Железнодорожный транспорт: актуальные задачи и инновации

Научно-технический журнал

Railway transport: Actual Tasks and innovations Scientific and technical journal

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