Transition Curves

During construction it is easy enough to lay a track in a transition configuration since there is no change in the method of attachment of rail to sleepers or baseboard. The difficulty lies in knowing the correct shape to create, i.e., just where to drive spikes, screws, etc. This is where mathematics comes in; and application of mathematical theory to track transitions is the subject matter of this section. The approach taken here is to start from the desirable characteristics of a railway vehicle on a track and see where that leads.

However, the conclusion is well established. The clothoid, aka Cornu spiral, aka Euler spiral, aka spiro, is not the only way of producing a transition curve; but it is the option of choice in modern times for road, rail, roller-coaster, and, probably, other transitions. This choice is made because the clothoid has characteristics nicely suited to these applications. The downside is the involvement of relatively advanced mathematics and significant computation. The latter is, of course, not a key issue when an electronic computer is available.

Some places to look for more information are given below. Much of what is written here has been gleaned from internet websites.

http://physics.gu.se/LISEBERG/eng/loop_pe.html
https://en.wikipedia.org/wiki/Euler_spiral
https://en.wikipedia.org/wiki/Fresnel_integral
http://www.mrol.com.au/Articles/Track/Cant_and_Transition_Design.aspx

There is no sideways force due to track curvature on a railway vehicle on a straight track. On a circular arc of radius $R$ the sideways force is the centripetal force $\frac{m<!--INVISIBLE\; TIMES-->v^2}{R}$, where $m$ is the mass of the vehicle, and $v$ is the velocity along the track. Thus, for constant vehicle mass and velocity, the centripetal force is proportional to the inverse of the radius: ${\mathrm{F}}_{cp}\propto <!--PROPORTIONAL\; TO-->\frac{1}{R}$. Furthermore, it is possible to refer to the curvature of a railway track, rather than its radius; the advantage being that the curvature $\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->$ of straight track is zero; and zero curvature may be more comfortable to think about than the inverse of an infinite radius. So ${\mathrm{F}}_{cp}\propto <!--PROPORTIONAL\; TO-->\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->$, where $\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->=\frac{1}{R}$.

Hence, the section of transition track under consideration, of length $L$, provides a transition between zero curvature ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_o$ at one end and some non-zero curvature ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_L$ at the other end. And, by doing so, this track provides a transition from zero sideways centripetal force on the railway vehicle to a maximum ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_L<!--INVISIBLE\; TIMES-->\mathrm{m}<!--INVISIBLE\; TIMES-->v^2$ at the other end. It is not required that the whole range of curvatures be used: by truncating the beginning of the track, i.e., by selecting a tail piece of the generated shape, the transition can be between two different non-zero curvatures, where the first is less than ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_L$. And, by travelling down the track in the reverse direction, the transition is an easement, instead of a tightening, of the curve.

For a small increment along the curve $\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->t$ in the first quadrant, the coordinates are $\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->x=\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->t.\cos <!--FUNCTION\; APPLICATION-->\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->\theta <!--GREEK\; SMALL\; LETTER\; THETA-->$ and $\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->y=\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->t.\sin <!--FUNCTION\; APPLICATION-->\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->\theta <!--GREEK\; SMALL\; LETTER\; THETA-->$, where $\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->\theta <!--GREEK\; SMALL\; LETTER\; THETA-->$ is the slope of $\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->t$. The slope is the tangent $\frac{\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->y}{\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->x}$, i.e., the tangent vector $[\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->x,\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->y]$ of the transition curve. Thus, arbitrary point $P_{\left(t\right)}=(<mi>x</mi>,\; <mi>y</mi>)$ is obtained by integrating along the contour of the transition curve:

With respect to curvature: for a circular arc of length $a$, radius $r$, and angle $\phi <!--GREEK\; SMALL\; LETTER\; PHI-->$ subtended by the arc at its centre, the relationship between these characteristics is given by $a=r<!--INVISIBLE\; TIMES-->\phi <!--GREEK\; SMALL\; LETTER\; PHI-->$. Then, applying this equation to an infinitesimal arc of the transition curve, and re-arranging, yields $\frac{1}{R}=\frac{\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->\theta <!--GREEK\; SMALL\; LETTER\; THETA-->}{\delta <!--GREEK\; SMALL\; LETTER\; DELTA-->t}$. Using equivalences from above, and taking the limit, yields the value of the transition curvature at the arbitrary point $P_{\left(t\right)}$ as ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_{f<!--FUNCTION\; APPLICATION-->\left(t\right)}=\frac{d\theta <!--GREEK\; SMALL\; LETTER\; THETA-->}{dt}=\frac{df<!--FUNCTION\; APPLICATION-->\left(t\right)}{dt}$.

If $f<!--FUNCTION\; APPLICATION-->\left(t\right)=t$ then ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_{f<!--FUNCTION\; APPLICATION-->\left(t\right)}=\frac{df<!--FUNCTION\; APPLICATION-->\left(t\right)}{dt}=1\forall <!--FOR\; ANY-->t$. Thus, the curvature is constant, the transition curve is a circular arc, and this does not satisfy the requirement of a smooth change from one curvature to another.

If $f<!--FUNCTION\; APPLICATION-->\left(t\right)=t^2$ then ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_{f<!--FUNCTION\; APPLICATION-->\left(t\right)}=\frac{df<!--FUNCTION\; APPLICATION-->\left(t\right)}{dt}=2<!--INVISIBLE\; TIMES-->t\forall <!--FOR\; ANY-->t$. Thus, the curvature is linear with respect to the length from the zero curvature end, and this does satisfy the desirable contour referred to earlier. This substitution yields the clothoid, commonly used for transition curves.

The curvature, ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_L$, of the transition curve at the non-origin end, matches the curvature of the curve to which it attaches, thus is known. Also known is the required length, $L$, of the section of track, i.e., the transition curve. A third requirement is is a units-balancing, numerically-normalizing, constant. This constant will be non-varying for any given curve, but otherwise will vary according to specific input values.

Since ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_{f<!--FUNCTION\; APPLICATION-->\left(t\right)}=2<!--INVISIBLE\; TIMES-->t\forall <!--FOR\; ANY-->t$ for the clothoid, at the non-origin end of the curve it follows that ${\kappa <!--GREEK\; SMALL\; LETTER\; KAPPA-->}_L=\frac{1}{R_L}=2<!--INVISIBLE\; TIMES-->L<!--INVISIBLE\; TIMES-->u^2$, where $u^2$ is the normalizing constant. Thus $u=\frac{1}{\sqrt{2<!--INVISIBLE\; TIMES-->R_L<!--INVISIBLE\; TIMES-->L}}$, and $1=2<!--INVISIBLE\; TIMES-->R_L<!--INVISIBLE\; TIMES-->L<!--INVISIBLE\; TIMES-->u^2$. In general at point $P_{\left(t\right)}$, $1=2<!--INVISIBLE\; TIMES-->(R_t<!--INVISIBLE\; TIMES-->u)<!--INVISIBLE\; TIMES-->(L_t<!--INVISIBLE\; TIMES-->u)=2<!--INVISIBLE\; TIMES-->R_N<!--INVISIBLE\; TIMES-->L_N$, where $N$ means normalized, $L_N$ is used to compute the values of the coordinates of $P_{\left(t\right)}$ from the integrals, and $R_N$ is the radius at the point, computationally unused. Once computed, the values of the coordinates of $P_{\left(t\right)}$ must be divided by the normalizing constant to restore the input units values.

A simple example net effect of the latter result is that, from an angular point of view, if a semi-circle has N constant curvature modules all of the same length, then a pseudo semi-circle can be formed with N-1 of those constant curvature modules together with a transition and an easement "book-ending" them. However, the effective diameter of the resultant pseudo semi-circle is larger than that of the constant curvature semi-circle due to the increased arc length, so it is not a simple plug-in replacement. Furthermore, N modules for a semi-circle implies that the corresponding circle has 2N modules, an even number. If a circle is formed by (2M + 1) modules, then a semi-circle cannot be formed from the modules.