CC BY
45
Инноватика - производству
47
ИННОВАТИКА - ПРОИЗВОДСТВУ
UDK 621.336.2
K. K. Kim, Ju. A. Antonov
Petersburg State Transport University
K. K. ^м, Ю. A. Aнтонов
Петербургский государственный университет путей сообщения
MODEL OF THE INTERACTION OF THE CONTACT WIRE WITH THE PANTOGRAPH МОДЕЛЬ ВЗАИМОДЕЙСТВИЯ ПАНТОГРАФНОГО ТОКОПРИЕМНИКА С КОНТАКТНЫМ ПРОВОДОМ
The high-speed running of electric trains demands increasing the quality of the current collection, which depends on the interaction of the catenary and the pantograph. The opportunity of building some models of interaction of the contact wire with the pantograph based of the boundary problem for the self-adjoint operators is shown in this work.
Высокоскоростное движение электропоездов диктует повышенные требования к качеству токосъема, которое определяется взаимодействием контактной подвески и пантографного токоприемника. Показана возможность построения модели взаимодействия пантографного токоприемника с контактным проводом, базирующейся на краевой задаче для самоспряженных операторов.
current collection, catenary, pantograph, contact wire, self-adjoint operator, elasticity. токосъем, контактная подвеска, пантограф, контактный провод, самосопряженный оператор, эластичность.
Introduction
The high-speed movement of an electric train requires improving the quality of current collection. The main problem is to describe the interaction of the catenary and the pantograph. A class of interaction models is offered in this work. It appears because even the simplest one-parameter is able to calculate the eigenfrequencies of the catenary.
Mathematical model
The basic assumptions used in this article are:
1. The contact wire oscillations take place only in the vertical plane.
2. The oscillation amplitude is small enough to neglect nonlinear effects.
3. The pantograph is considered as a source of a constant force.
4. The train speed is constant.
5. The strength tension of the contact wire is constant along its length.
6. The model may be formulated as a boundary problem for a self-adjoint operator.
We use the next model equation for the mathematical description of the process:
d Y^t) + LY(x, t) = — 5(x - vt) (1)
dt2 pS
where Y(x,t) is the shift of the contact wire from its state of rest at the point x at the moment t, L is a self-adjoint differential operator, which does not depend on t; p, S, F, v are
ISSN 1815-588Х. Известия ПГУПС
2012/2
48
Инноватика - производству
respectively corresponding to the density, the cross section area of the contact wire, the acting pantograph force and the moving train speed.
Let’s consider that the static interaction is described in the following way:
~ F
Ly{x x0) = —5(x - x0X PS
where y(x, x0) is the shift of the contact wire from its state of rest at the point x at that moment when the pantograph is at the point x0. It is known from the theory of differential equations that any self-adjoint operator L possesses the complete system of orthonormal eigenfunctions {yn(x)} in L2 [2]. These functions satisfy the equation Lyn = Xnyn and corresponding boundary conditions. Therefore any function z(x) can be expanded into the convergent series using {yn(x)} as a basis:
да
Z(x) = i ВпУn (x).
n=1
Let’s assume that the physical model of the real process is such a process when the considered functions are sufficiently smooth and satisfy all
the necessary conditions. The decomposition coX 2
efficients can be found as: Bn = J z(x)yn (x)dx,
Xi
where X1 and X2 are the definitional domain boundaries of L. The using of the series instead of the function leads to the simplification of the way to find the procedure. Let’s apply this method to the static interaction equation. Expanding the function y (x, x0) and the right quotient of the equation into series, you can get:
да
У(^ X0) = i Rn (X0) У n (x)
n =1
and
F
Rn (X0)X n =—j'V n (x0).
PS
Now you can determine the elasticity of the contact wire using the attributes of the model operator L .
It follows from the definition that the elasti-
. . , ч y (x, x) „
city is n( x) = —---. By using it, you can get:
F
( \ 1 y n (x)
n( x) = -^L~n—.
pO n X n
Integrating the preceding result you can calculate the average elasticity:
n =
—x—.
pSLr Xn
The same method should be applied to the model equation. Let’s find the solution as:
да
Y(x, t) = ХФп (t) Уn(x\
n =1
where the functions фи(^ are used as unknown decomposition coefficients. Let’s substitute this series into themodel equation. Taking into account that Ly n = X n y n, you can get
F
фП (t)+хпФп (t) = ^Уп (vt) for functions ф„(0. PS
This equation has the well-known solution:
Фп (t) = P1n (t) sinWXnt) + P2n (t) cos C/Lt),
where P1n (t)
t
Jy n (vT)cos^/XnT)d т,
0
P2n(t) = -f J Уп (vt) sin^lKT)dT f = F.
Vхn 0 pS
Then, using this result you can get the next formulae:
F_
PS
5(x - x0) = X Dn (x0) У n (x),
n =1
Y (x, t) = f x
n =1
where the coefficients Dn (x0) can be easily calculated from the last equation by integrating:
Dn (x0)
F_
PS
y n(x0).
We can use the orthogonality of the functions yn(x) to get the equation for the values Rn(x0):
x J yn (vt) sin^JXn(t - T))dт
0
for the value of the contact wire shift.
In order to use the above results it’s necessary to choose the explicit form of the model operator L .
2012/2
Proceedings of Petersburg Transport University
Инноватика - производству
49
Mind that the average elasticity must be nearly constant along the anchor section. To obtain such a result from the model you should suppose that forces act on the contact wire. Let’s consider that these forces are arranged periodically along the anchor section and that their magnitudes are proportional to the shift of the contact wire from its state of rest at the point of force application. In this article we shall consider a simple one-parameter model of interaction of the catenary and the pantograph:
L = -—^2-X ^ 8( X -x.),
-S dx i=i pS
As the left part has b = 0 the value of a (the scaling coefficient) can be chosen arbitrary, for example an = 1. At the same time on the right part we have an sin(kn 2L) + bn cos(kn 2L) = 0. Moreover, at the point of the force application the following two conditions should be satisfied:
sin(knL) = an sin(knL) + bn cos(knL)
and
kn (an - 1)cos(knL) --knbn sin(knL) = — sin(knL).
where x., s., N, Tare respectively the coordinates of the points of force application, the spring constants, the numbers of acting forces on the anchor section, the tension of the contact wire. This is an obvious self-adjoint operator. Therefore, you can use the above described method. The model equation for the anchor section is expressed by the formulae:
The first equation expresses the wire continuity at the point x = L. The second equation is the jump condition for the derivative at the point of applying the force. You can state that this model equation has a virtue of integration. You should eliminate the coefficients a
n
and bn from the last formula to get the equation for the values k :
n
d2Y(x, t) - — d2Y(x, t) -dt2 pS dx2
Ns
-X-Ь§( x - x)Y (x, t) = f§( x - vtX
i=1 -S
with the boundary conditions Y(0,t) = Y(L,t) = 0.
To understand the method used let’s consider the following simplest problem: we have a fastened spring in the middle of the strained wire with the fixed ends. It is necessary to find out the eigenfrequencies of this system. Let’s denote the spring constant as s, and the length of the wire as 2L. The wire is divided into two uniform parts by the spring. The oscillations at each part are described by the ordinary wave equation whose solution can be represented in the next form:
an sin(knx) + bn cos(knx),
where kn
, and c V c
is the phase
velocity.
The coefficients should be determined from the boundary conditions. The solution must be equal to zero at the points x = 0 and x = 2L.
2 cos(knL) + ~^~ sin(knL) = °.
knT
This equation can be easily solved numerically. After that, the sets of the eigenfrequen-
ckn
cies may be found as: vn =^~. Increasing
the number of strings leads to the significant complication of the equation for the values kn. However, you can write it for any given quantity of strings. It means that, as a result, we get a power equation. Moreover, the exponent of the equation is equal to the number of strings and the argument of the equation is
s
2 cos(knL) + —— sin(knL).
knT
As a result, the problem of finding the values kn leads to the case of two rather simple numeric tasks: searching the set of the roots {za} of a polynomial and solving the following equation:
2 cos(knL) + -^ sin(knL) = za.
knT
The magnitude of s greatly influences on the values of eigenfrequencies. Within the
ISSN 1815-588Х. Известия ПГУПС
2012/2
50
Инноватика - производству
limits of the model considered parameter s determines the properties of the catenary as a whole. It should be chosen so that the average elasticity calculated by virtue of the suggested
method, i. e. Л = _~ X"1, agrees closely
pSL n An
with the experimental value. The described method was used (only as an illustration) to the section, which consists of two segments of the catenary КС-200 the parameters of which are given in the table shown below.
the number of the strings 27
the average elasticity 0,37 mm/N
the length of the section 130 m
the tension of the contact wire 12000 N
the phase velocity 106 m/s
As it was stated above we have to find out the set of the roots {za} of a polynominal equation. The coefficients of this equation are determined only by the number of strings. So, in the considered case this equation looks like:
z27 - 26z25 + 300z23 - 2024z21 +
+ 8855z19 - 26 334z17 + 54 264z15 -- 77 520z13 + 75 582z11 - 48 620z9 +
+ 19 448z7 - 4368z5 + 455z2 3 -14z = 0.
After defining the initial value of the parameter s the set of equations
2 cos(knL) + ~^~ sin(knL) = za
knT
should be solved to get the values kn and then to calculate the average elasticity. You should repeat this procedure until you achieve the coincidence between the theory and the experiment. In this way, the value s = 500 N/m is calculated and the set of eigenfrequencies of the catenary: Vj=1,64 Hz, v2=1,78 Hz, v3=2,01 Hz and etc. is computed.
Conclusion
Thus, the formula for the wave arising in the catenary and for the elasticity as a series of the eigenfunctions of the boundary problem for the model self-adjoint operator have been received in this work. Within the framework of the simple one-parameter model of interaction of the catenary and the pantograph you get the equation for the eigenfrequencies of the catenary. The method of its solution is suggested. On the base of the obtained results you may find out the factors acting on the quality of the current collection.
References
1. Kurs vyshey matematiki / V. I. Smirnov. -M. : Nauka, 1958. - 881 s.
2. Nonlinear vibrations in mechanical and electrical systems / J. J. Stoker. - N. Y., 1950. - 256 p.
УДК 656.02
В. А. Кудрявцев, В. Л. Белозёров
Петербургский государственный университет путей сообщения
ИНТЕНСИФИКАЦИЯ РАБОТЫ СОСТАВОВ ПАССАЖИРСКИХ ПОЕЗДОВ НА ОСНОВЕ СОВМЕЩЕНИЯ ИХ ОБОРОТА
Излагается научно-методическое обоснование необходимости совмещения оборота пассажирских составов, что позволяет осваивать пассажиропоток меньшим числом составов. Определены
2012/2
Proceedings of Petersburg Transport University
