Ìàòåìàòèêà /
4.Ïðèêëàäíàÿ ìàòåìàòèêà.
Glebov V.V., Danilova T.G., Skachkov V. K.
The College of Railway Transport,
The Ural State University of Railway Transport,
The city of Yekaterinburg.
Clothoids
in railways.
The modern railroads have the irregular geometric
shape unlike the first ones which consisted of the straight sections connected
by circular arcs. As time passed by, new technologies were developed and the
speed of trains increased, a great risk
of coming off the rails appeared.
To increase traffic safety, to improve working
conditions, the railway track should be smooth and with the clear direction for
a big distance, the plan and a profile of the way should be harmoniously
combined with an ambient landscape.
For the solution of objectives it is necessary to
consult physics, in particular to centrifugal force which formula is given as
follows:
where
F = centrifugal force,
m = mass,
v = speed,
r = radius.
From the formula we can see that the more the mass of
the train or its speed, the more centrifugal force. Accordingly to reduce this
force it is necessary either to reduce the mass of the train (but it can't be
made with modern traffic) or to reduce speed (it can't be also considered as a
variant of the solution of this problem).
On the other hand, there is a curvature radius which
is situated in a denominator of the given formula. It means that if we increase
radius, then it is possible to reduce centrifugal force. It is clear: radius of
curvature of a straight line is indefinite, so centrifugal force is equal to
zero in case of moveming on a straight line. Increasing of the radius of
curvature seems to be the only possible way of solving the problem, but this
method isn't ideal, it has its own disadvantages. According to such planning of
a railway track direct track sections will become shorter, and trains will pass
from one turn into another. [1]
Introduction of an ease curve between a straight line
and a circle is necessary, at the same time it is necessary that curvature
radius smoothly decreased from infinity for a straight line to circle radius.
According to the formula, centrifugal force will change smoothly, and not sharp
upon such transition. [2]
As a result of researches, the most successful way of
the movement has been recognized as: on a straight line — a clothoid — a circle
— a clothoid — a straight line. According such traffic pattern centrifugal
force changes gradually, the turn is smoothly, without breakthroughs and danger
of a coming off the rails. The main feature of the clothoid is that its
curvature is directly proportional to the distance. The section of the road, a
railway track where along with a direct and circular curve the clothoid is used
as an element of the plan, is called clothoidal.
The study of the clothoid was first proposed as a problem of
elasticity by James Bernoulli in 1694. In 1744 this problem was solved by the
mathematician and physicist Leonard Euler who described several properties of
the curve. Around 1818, french physicist Augustin Fresnel rediscovered clothoid
while studing a problem of light diffracting, and with the help of integrals
received the parameterization of this curve, equivalent to the Euler’s
parameterization. [3] In 1874, the French physicist Marie Alfred Cornu used
this expression to plot the curve accurately. And later, in 1890, American
engineer Arthur Talbot also opened clothoid when he was looking for the curve
of transition for the railroads. Clothoid is also known as the spiral of Cornu
or the Euler spiral. Clothoid is used eighter on tracks and on roller coasters. According to the modern
requirements of the International Ski Federation, the line of jump landing should consist such a curve as a clothoid.
[4]
The
introduction of clothoid transition curves together with the maximum smoothness
of the railway track provides a number of advantages such as improving traffic
conditions of trains, especially at night; increasing safety due to increasing
of visibility distance; reducing the volume of earthworks due to the best fit
of the rail track to relief; the comfort of the passengers during the trip.
References:
1.
V.F. Babkov. Modern
highways. M, Avtotransizdat,1961.
2.
S.A. Treskinskij, I. G.
Khudyakova. The physical basis of clothoidal tracing. M, Avtomobil’nye dorogi, ¹ 5, 1963.
3.
Andrew D. Swedberg.
Safer ski jumps: design of landing surfaces and clothoidal in-run transitions.
2010.
4.
R. Levien. The Euler
Spiral: A mathematical history. University of California at Berkeley. 2008