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(Basics of) Railway Curve Kinematics

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A brief description of basic kinematic parameters that are in use in railway horizontal alignment calculations.
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(Basicsof)RailwayCurveKinematics
Robert Wojtczak,
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Type of Horizontal Alignment Elements
Straight (Tangent)
Transition Curve
Circular Curve
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Type of Transition Curves
Cubic Parabola
• Clothoid
Bloss Curve
Cosinusoidal (Japanese)
Sinusoidal (Klein)
Bi-quadratic Parabola
10-chord AREMA
Wiener Bogen
• Others…
Most commonly used transition curve in railway industry is Clothoid.
This type of curve can be defined by parameter A:
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Horizontal Curve with Clothoidal Transition
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Cant (Superelevation)
Cant (C) is applied (generally) to:
Circular curves (full cant value)
Transition curves (between 0 and full cant value)
Cant is measured in mm and can be (depending on standards) between 0 and 180 mm.
For cubic parabola and clothoid the cant is in the form of linear function.
The part of track where Cant changes is called Cant Ramp.
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Why we add Cant
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Cant Deficiency
Curve equilibrium formula for standard gauge:
Cant Deficiency (D) can be calculated from difference between Equilibrium Cant (Ceq)
and Applied Cant (C). If the result is negative then we call it Cant Excess.
Usually, the limit for Cant Deficiency is 100 mm.
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Rate of Change of Cant
The rate in which Cant changes in time is called Rate of Change of Cant (RCC).
RCC can be felt by a passenger as vertical speed along the cant ramp.
Limits for this parameter are usually as follows: 28 mm/s (normal), 35 mm/s
(max), 55 mm/s (abs max).
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Rate of Change of Cant Deficiency
The rate in which Cant Deficiency changes in time is called Rate of Change of
Cant Deficiency (RCD).
RCD can be felt by a passenger as a change in horizontal pull experienced.
Limits for this parameter are usually as follows: 28 mm/s (normal), 35 mm/s
(max), 55 mm/s (abs max).
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Alternative Measures
There are some alternative parameters in use that have direct connections to D,
RCC and RCD:
Unbalanced Lateral Acceleration (a)
<equivalent of Cant Def.>
Vehicle Roll (φ)
<equivalent of RCC>
Rate of Change of Unb. Lat. Acceleration (ψ)
(Jerk)
<equivalent of RCD>
(g – Earth’s gravity, s – rail centers)
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Curve Calculations Example
Speed
km/h
Radius
m
Cant
mm
Trans
m
Min
Trans
Joins
next?
Def
mm
RCC
mm/s
RCD
mm/s CG,1:_
160 3000 65 105 103.2 35.7 27.5 15.1 1615.0
120 1000 100 120 119.0 69.9 27.8 19.4 1200.0
80 380 100 80 79.4 98.7 27.8 27.4 800.0
CURVECALCULATIONS
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Curve Graphs Example
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Q & A
Thank you. Questions?
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