Equivalent Conicity

What Equivalent Conicity Is

A railway wheelset is self-steering because its wheels are conical: shifted sideways, the wheel on one side rolls on a larger radius than the wheel on the other side, and the wheelset steers back to the center. How strongly it does so is the equivalent conicity, tan γe. Too low and the vehicle wanders in the track; too high and the wheelset hunts, a sinusoidal oscillation that shakes the vehicle and wears the rails. Because real wheels and rails are not cones, tan γe depends on the actual shape of both profiles, on the gauge and on the rail inclination, and it changes as the rails wear. EN 15302 defines how it is calculated from the two profiles, and infrastructure managers use it to decide where the rails need grinding or reprofiling.

Our Implementation

KZV has its own implementation of EN 15302. It is verified against every test case published in the standard (Annexes I to K): for each pair of wheel and rail profiles, the contact points, the rolling radius difference Δr and the tan γe curve match the reference results within the tolerances the standard sets. The calculation uses the two-step integration method, which is the more robust of the methods the standard allows: it gives a stable result for worn profiles with sudden changes in the contact geometry, where linear regression is unreliable.

The implementation is efficient: it processes thousands of rail profiles per second on ordinary consumer hardware. That is far more than the profile scanners deliver, so the equivalent conicity of the measured track is calculated during the run rather than in the office afterwards.

How It Is Delivered

• Real time in Krab OnBoard: the conicity of the track under the trolley or the measuring vehicle, calculated during the run from the gauge and the two rail profiles the trolley measures together (Krab OnBoard, version 2.0 and 3.0)
• In your own software: the calculation core is integrated into the customer's evaluation, monitoring or asset-management software on request
• Interactive program: a stand-alone program for Windows, Linux or the web browser to load measured profiles and explore the contact points, the rolling radius difference, tan γe and its sensitivity to measurement noise

Contact Geometry in Detail

The interactive program shows what goes into the number. Wheel profiles come from a CSV file, nominal or measured; rail profiles and gauge from the measurement. For a chosen wheel and rail pair, the contact points for every lateral displacement are drawn on both profiles, and the resulting rolling radius difference Δr is plotted against the displacement. The tan γe curve is shown against the amplitude ŷ, with the two-step integration result next to the linear regression, the expected curve of the standard's test case and its tolerance bounds. The smooth assessment distorts the profiles with random measurement noise of a chosen magnitude and repeats the calculation many times, which shows how much a given scanner accuracy can move the result. The wheel and rail profiles of the standard's test cases are built in.

From Measured Profiles to Conicity

• Measure: gauge and both rail profiles in one run with KRAB Heavy, KRAB S-LIGHT or TURTLE fitted with profile scanners
• Calculate: tan γe in real time in Krab OnBoard, in your own software or in the interactive program
• Decide: where the rails need grinding or reprofiling, based on the conicity along the track