Equivalent conicity

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In railways , the equivalent conicity of the wheel - rail contact determines the running properties. It results from the geometry of the running surface of the wheels and the surface of the rail head. It is defined as the inclination of a conical wheel profile rolling on sharp edges, which would result in the same wavelength of the sinusoidal course , and is a function of the amplitude .

Conical wheel profile - original profile

Tapered wheel profile new condition / worn

The original profile of the railway wheel was a truncated cone (conical course of the profile). The conicity of this original profile is the tangent of half the opening angle of the cone. Based on this original profile, various wheel profiles have developed whose profile is non-linear. The illustration shows the conical original profile when new. The profile shape of the tapered profile when worn is shown in red.

For worn or non-tapered profiles, the rolling radius difference is a function of the lateral displacement of the wheel set .

Rolling radius difference

Due to the conical inclination, the right wheel, which is just running against the rail in the illustration, has a larger radius than the left wheel, which runs further inside. The figure shows the transverse displacement , the rolling radii for transverse displacement zero and the rolling radii and for a given displacement as well as the cone inclination .

The following relationships apply to wheels with a conical running surface:

and

This gives the rolling radius difference:

whereby when new, according to regulations, is usually 1:40 or 1:20.

Purely tapered wheel profiles have the advantage of high running stability, but the following disadvantages:

  • the running surface of the wheel always touches the rail in the same place - the result is a narrow driving mirror on the rail
  • the narrow contact surface leads to a high level of wear on the wheel and the rail; the wheels therefore often have to be turned off because the driving characteristics deteriorate considerably as the wear increases. The shorter service life of the rails is associated with high maintenance costs

Wear profile

Wear profile ORE 1002

In order to avoid the disadvantages resulting from the wear and tear of the conical wheel profiles, a search was made for dimensionally stable, geometric wheel-rail pairings that also had sufficient running stability. One of these developed wheel profiles is the so-called UIC-ORE standard profile S1002. With this profile, the running surface no longer runs linearly, but as a continuous curve which is specified in xy coordinates (see figure on the right - the coordinate tables are not specified here).

The linear law for the conicity as with the original profile no longer applies. The rolling radius difference becomes a non-linear function . These types of profile are known as wear profiles because the profile of the wheel when new is designed in the same way as it would appear after a high mileage. One can then assume that the worn wheel profile has reached a stable state in which it shows only slight further wear.

Although wear profiles have the advantage of low wear, they have the following disadvantage:

  • poor running behavior, which is characterized by the equivalent taper (the higher the equivalent taper, the worse the running behavior), which is influenced by the following parameters:
    • Gauge (the narrower the gauge, the higher the equivalent taper)
    • Rail inclination, whereby rail inclinations 1:20 (e.g. used in France) have significantly lower equivalent conicities than rail inclinations 1:40 (e.g. used in Austria)
    • Rail profile
    • Wheel profile

Since the wheels of the railway are rigidly connected to the axles, the vehicle runs in waves ( sinusoidal ) on the track, depending on the difference in rolling radii . If the right wheel runs z. B. to the right, then its running radius increases (over the cone slope), the left wheel then runs on a smaller running radius. Because the wheels are rigidly connected, the right wheel makes more travel (over the larger circumference) and begins to steer back towards the center. This process is repeated in interplay. In this way, the undulating sine wave of the drive results.

Klingel's formula

A formula for the frequency of this wave-like movement of the free wheel set was derived from building officer Johannes Klingel from Karlsruhe in 1883:

, With

It means:

... equivalent conicity

... rolling radii

... medium rolling radius

... support spacing (usually 1500 mm for standard gauge)

... running frequency

... speed

From the above relationship it can be seen that the running frequency increases with increasing equivalent conicity, that is to say that the vehicle is stressed with a high excitation frequency from the sinusoidal running of the wheelset. The critical speed at which the vehicle runs stably decreases. For high travel speeds, the contact geometry between the wheel-rail must therefore be selected in such a way that small equivalent conicities occur.

The following are guidelines:

Type of bogies Critical speed in km / h
0.2 old 120
0.2 modern 160
0.1 modern 200-250
Difference in rolling radius via curve radius

When it comes to running behavior, a distinction must be made between sheet runnability and running quality in straight lines.

For stable running in the curve, wheel profiles should be selected that have the largest possible rolling radius difference on the existing rail profiles. This means that the vehicle can run on the outer wheel of the curve on the large rolling radius and on the inner wheel on the small rolling radius with as little slip and vibration as possible. A large equivalent taper is therefore an advantage in the arc run. The two curves shown on the right apply to different wheel diameters .

For stable running in a straight line, however, the frequency of the sinusoidal running of the gear sets should be as small as possible - which requires the smallest possible equivalent conicity.

Influencing parameters

Rail inclination

Dependence of the equivalent conicity on the inclination of the rail installation

Different rail inclines are used in different countries. That is, the rail is mounted on the sleepers so that it is inclined inward. The most common are the inclinations 1:40 and 1:20, but the rail inclination 1:30 is also used in isolated cases.

As can be seen in the adjacent picture, a rail inclination of 1:20 results in particularly low tapers that are largely independent of the gauge. This rail inclination is therefore mainly used on high-speed routes (e.g. in France on the LGV lines). The disadvantage of the rail inclination of 1:20 is a less favorable contact geometry between wheel and rail, which is why the wheels have to be turned more often due to the increased wear. So that the vehicles in high-speed traffic (v ≥ 250 km / h) are suitable for the high driving speeds, i. d. Usually the wheel sets are rigidly coupled and roll dampers are built up.

Wheel profile

As already stated above, the wheel profile naturally has a decisive influence on the running behavior of the railway vehicle. Wear profiles are used so that the maintenance costs of the wheelsets can be kept low. Although these have a higher mileage before they have to be re-profiled, this advantage is bought at the price of poorer running behavior, characterized by a high equivalent taper.

At DB AG z. B. used a wheel profile that corresponds to the UIC-ORE S1002 wear profile. When new, this has an equivalent taper of 0.17 with the UIC60 rail pairing, with a nominal gauge of 1435 mm and a rail inclination of 1:40. This pairing no longer meets the high requirements for running quality at speeds over 250 km / h. One solution to this problem lies in the special design of the rail profile used.

Rail profile

One of the partners in the wheel-rail pairing is the rail. The rail profile thus also has a decisive influence on the equivalent conicity that results. At DB AG, the rail profile was therefore further developed at the end of the 1990s, particularly with a view to the construction of the high-speed line Cologne – Rhine / Main with a planned operating speed of 300 km / h. The result of these efforts was the rail profile 60E2, which has been the standard rail profile of DB AG since 2000. It can also be easily produced during operation by profiling grinding.

Gauge

Equivalent taper depending on the gauge

The track width has a significant influence on the equivalent conicity - in particular, track narrowings below 1432 mm have a strong effect on the equivalent conicity. Lane narrowing must therefore be eliminated. These can be remedied by appropriate profiling grinding of the rails. Another option is to adjust the rail fastening. In any case, the cost of this maintenance is high. The illustration opposite shows the progressive increase in the equivalent conicity in the case of track narrowing.

Critical speed of railway vehicles

Limit values ​​for driving safety and road stress
Unstable running bogie without SD roll damper

The equivalent conicity is an important parameter for describing the running behavior of a railway vehicle. Stability studies, based only on the linearized equivalent conicity, are not sufficient for assessing the critical speed of a vehicle (the speed up to which the vehicle is still running in a stable manner). They represent a lower limit. For an exact calculation, the non-linear dynamics must also be taken into account and examined. The figure shows an overview of the limit values ​​with regard to the driving safety of railway vehicles and those for the stress on the route through the rolling material. The measurement curves in the figure below show as an example a bogie DG1 running unstably due to the high conicity and the missing roll damper. Because of the increase in the equivalent conicity shown in the figure below, it exhibits high rolling, as the measured accelerations show.

literature

  • Anton Nefzger: Geometry of the contact between wheelset and track. In: ETR-Eisenbahntechnische Rundschau. Volume 3, 1974, pp. 113-122.
  • Anton Nefzger: Running dynamics in cross-border high-speed traffic. In: EI railway engineer. 42, 1991, p. 106ff.
  • Reinhard Walenta, Andreas Haigermoser: Contact geometry of wheel and rail - New methods for investigation and optimization. In: ZEV + DET Glaser's annals. 121, 1997, No. 2/3, pp. 245-254.
  • Bernd Bergander, Günter Derndl, Anton Nefzger, Dirk Nicklisch: The development of wheel and rail profiles . In: ZEVrail Glasers Annalen. 127, 2003, h.10; Pp. 482-493.
  • Hans True: On the equivalent conicity. In: ZEVrail Glasers Annalen. 131, Proceedings SFT Graz 2007, pp. 290–298.
  • Bernd Bergander, Peter Meinke, Anton Nefzger: Interaction between vehicle and track - technical principles and practical application. In: Railway engineer calendar. 97, Association of German Railway Engineers EV VDEI, pp. 135–159.
  • Roland Müller: The problem of the wheel / rail contact geometry. In: ZEV + DET Glaser's annals. 118, 1995, No. 3, pp. 86-99.
  • Walter Rode: History of the Stability Criteria. In: EI railway engineer. (55) 1/2004, pp. 35-39.
  • Bernhard Lichtberger: Track manual. Tetzlaff Verlag, Hamburg 2003, ISBN 3-87814-803-8 .
  • DIN EN 15302: 2011-01: Railway applications - Method for determining the equivalent conicity ; German version

Individual evidence

  1. Klaus Knothe, Sebastian Stichel: Rail Vehicle Dynamics . (VDI book), 2003, ISBN 3-540-43429-1 , Google Books .