The track geometry is governed by numbers and all the time we are using them to design and check our work.
The geometry of the non-linear transition curves is also governed by important numbers, useful to know in the alignment design when these transitions are used:
– Cosine (and its close approximation: the half-sine or Japanese transition)
– Bi-quadratic parabola (Helmert or Schramm)
– Sine (Klein)
Two important sets of numbers used in the high speed alignment design are:
– qN – the cant parameter factor for non-linear transitions. This defines the transition cant and curvature parameter proportional values compared to the clothoid.
For example the rate of change of cant deficiency (RgD) for Cosine is
RgD (Cosine) = 1.5708 • RgD (Clothoid)
This is true if both transitions, Cosine and Clothoid, have the same length.
This factor is used for establishing the length of the transition and to compute all the cant related parameters.
– sN – the shift factor for non-linear transitions. This defines the length of the transition compared to the clothoid length (L) to provide the same shift of the circular arc
For example the length of the Cosine transition compared with the length of the Clothoid is
L (Cosine) ≈ 1.3265 • L (Clothoid)
to provide the same shift of the circular curve.
This factor is used in the alignment design especially to allow the design of the initial alignment using clothoids and considering the insertion of the non-linear transition in the final alignment without any change of the rest of the alignment components, straights or circular arcs. This method is used when designing different variants of an alignment in very constrained areas and especially when designing reverse transition.
The factors sN are indicative, still providing a below 1% precision for length. If a more precise length value is desired the shift correlation should be based on a precise shift equation definition both for Clothoid and for referred non-linear transition.
For the reverse transition the factors presented here are true if the reverse non-linear transition is defined as one transition arc and not as two separate transitions going from one radius to the reverse point of null curvature and forward from it to the second radius. This is because, for all non-linear transitions, the reverse point of null curvature is not placed in the point of length balance, as it happens for Clothoid, where this rule is applied:
L1 • R1 = L2 • R2
This essential difference between the linear and non-linear transitions will be presented in a future post.
EN 13803-1 (2010) Railway Applications – Track – Track alignment parameters – Track gauges 1435 and wider – Part 1: Plain Line