3abdaf3b-77c6-4b66-901f-0d524c4a5f03 7 Martin Kraska 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

Transition Curve

The task is to design the transition curve for a railway track changing from curvature \[REGION[871a7486247b4ecb97eaf731ca4f34c2]]\ to a curvature \[REGION[a1d58ed0bf1e48c88ab3a0d418a17a4b]]\ with limited rate of curvature change \[REGION[53c5242a08ad42909e6886fbebce5d1d]]\ and limited rate of transitioning to that rate of change (which means limiting \[REGION[6da27aef1bfd45488a5d94fbdcdebe81]]\.Given these limits, the curve has to be as short as possible. If we hadn't the limitation of \[REGION[a8bb73a17ce04600b08da19492136749]]\, the solution would be a clothoide (curve of linear \[REGION[43c9b4d425624ffd874f4dacebc67c23]]\).

Background: In curves, the outer rail is superelevated in order to align the roll angle with the resultant of centrifugal force and weight for minimum transverse forces. For a given velocity of the train, \[REGION[7df8d564defb431eabd667f4652c40cd]]\ limits the angular velocity and \[REGION[1f4726f449ac4c15be453d970110ba57]]\ limits the angular acceleration of this adjustment.

We first generate the curvature as function of position (arc length) \[REGION[46220b73c27848e5aa8d8504fba09568]]\ and then integrate the the curve using the Runge-Kutta method.

κ.0

κ.0 Δκ +

κ'

κ''

κ''

s κ

κ'

κ''

s κ

κ.0 0 :

Initial curvature

Δκ 1250 m * / :

Total change of curvature

Maximum rate of curvature change

κ'.max 101 km 2^/ * :

Maximum second derivative (spatial acceleration)

κ''.max 1001 km 3^/ * :

We apply the formulas for motion with constant acceleration to the curvature as a function of arc length.

Phase 1: Acceleration to \[REGION[c378a6b4d97d470f8f99bdae83a28d01]]\

κ'.max

Δs.E κ'.max κ''.max / : 100 m *

Arc length

Δκ.E 12 / κ''.max * Δs.E 2^* : 1 km / 0.5000

Change of curvature

Phase 3: Deceleration to \[REGION[e148417742e9495c8915ef577626643d]]\

κ' 0 ≡

Arc length and change of curvature are the same as in

Phase 2: Constant rate of curvature change

Remaining change of curvature

Δκ.L Δκ 2 Δκ.E * - : 1 km / 3

Δs.L Δκ.L κ'.max / : 300 m *

Arc length

Total length

Δs 2 Δs.E * Δs.L + : 500 m *

Definition and plot of \[REGION[cf1405763a2040a99876ff1ce215b7a0]]\

s κ

s κ κ.0 s 0 < κ.0 12 / κ''.max * s 2^* + s 0 Δs.E lele κ.0 Δκ.E + κ'.max s Δs.E - (* + s Δs.E Δs.E Δs.L + ltle κ.0 Δκ + 12 / κ''.max * s Δs - (2^* - s Δs.E Δs.L + Δs ltle κ.0 Δκ + cases :

0 κ 0.000

Δs.E κ Δκ 0.1250

Δs.E Δs.L + κ Δκ 0.8750

2 Δs.E * Δs.L + κ Δκ 1.000

Integration. The curve starts at the origin in horizontal direction.

xaxis XYPlot'XAxis : \007B\xaxis\007D\\0027\DecimalPlaces 3 : \007B\xaxis\007D\\0027\Max 500 : \007B\xaxis\007D\\0027\Min 0 : \007B\xaxis\007D\\0027\Tick 100 : \007B\xaxis\007D\\0027\NumberFormat General : \007B\xaxis\007D\\0027\Visible true : 71 line

yaxis XYPlot'YAxis : \007B\yaxis\007D\\0027\DecimalPlaces 3 : \007B\yaxis\007D\\0027\Max 300 : \007B\yaxis\007D\\0027\Min 0 : \007B\yaxis\007D\\0027\Tick 50 : \007B\yaxis\007D\\0027\NumberFormat General : \007B\yaxis\007D\\0027\Visible true : 71 line

Number of steps

n 100 :

Differential equations and initial conditions

Location and direction at the end of the transition

D s φ s x s y 3 1 sys Δs m 1 : 1 1 mat n 2 1 line rkfixed :

s φ' s κ ≡

0 φ 0 ≡

φ.1 Δs φ : ° 57.3

s x' s φ cos ≡

0 x 0 ≡

P.1 Δs x Δs y 12 mat : 45614312 mat

s y' s φ sin ≡

0 y 0 ≡

xaxis XYPlotk'XAxis : \007B\xaxis\007D\\0027\DecimalPlaces 3 : \007B\xaxis\007D\\0027\Max Δs : \007B\xaxis\007D\\0027\Min 0 : \007B\xaxis\007D\\0027\Tick 100 : \007B\xaxis\007D\\0027\NumberFormat General : \007B\xaxis\007D\\0027\Visible true : 71 line

yaxis XYPlotk'YAxis : \007B\yaxis\007D\\0027\DecimalPlaces 3 : \007B\yaxis\007D\\0027\Max 1 : \007B\yaxis\007D\\0027\Min 0 : \007B\yaxis\007D\\0027\Tick 0.2 : \007B\yaxis\007D\\0027\NumberFormat General : \007B\yaxis\007D\\0027\Visible true : 71 line

We overwrite the unit locally in the integration function because it is not unit-aware.

Diagram

Center of tangent circle at the end of the transition curve

x.c P.1 1 el 1 Δκ m * / φ.1 sin * - : 245

y.c P.1 2 el 1 Δκ m * / φ.1 cos * + : 278

D 3 col D 4 col augment P.1 o augment x.c y.c + C 2 1 mat 5 10 2 1 mat red augment x x.c - ( 2 ^ y y.c - ( 2 ^ + ( 1 Δκ m * ( 2 ^ / - 4 1 sys s m * κ Δκ / 1 1 sys