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Transition Curves

The problem that is addressed by transition curves is the sudden change in centripetal acceleration as a moving vehicle enters a fixed radius curve. At slow speeds this is rather trivial; but as speed increases it becomes a significant jerk, unpleasant for people, stressful for hardware. A transition curve eliminates the jerk by providing a continuous change from one centripetal acceleration to another. In the model and miniature railway arena the issue mainly is the visual smoothness of a train as it enters or exits a curve from or to a straight track. But smoothness, or otherwise, affects realistic operation significantly.

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.

Requirements
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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 m v2 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: F c p 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 κ of straight track is zero; and zero curvature may be more comfortable to think about than the inverse of an infinite radius. So F c p κ , where κ = 1 R .

Hence, the section of transition track under consideration, of length L, provides a transition between zero curvature κ o at one end and some non-zero curvature κ 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 κ L m v2 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 κ L . And, by travelling down the track in the reverse direction, the transition is an easement, instead of a tightening, of the curve.

Although the end points of this track section have specified curvatures, the contour, i.e., the way the curvature varies down the length of the section, has not been specified. At a practical level, this is fairly simple to handle; for example, having the curvature linear with respect to its distance from the zero curvature end is one obvious desirable contour. Thus at L 4 the curvature would be one quarter of the maximum, at halfway the curvature would be half the maximum, etc. However, at this point in this description the contour will be left undecided by specifying it simply as some function f ( t ) of the distance t from the zero curvature end; i.e., the curvature at some arbitrary point, distance t from the zero curvature end, is κ f ( t ) , and the centripetal force is given by F c p ( t ) = κ f ( t ) m v2 .

Coordinates
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What is required for track construction is a set of Cartesian coordinates specifying the transition curve shape. Thus, in the equations that follow, the origin of the transition curve is at the origin of Cartesian rectangular coordinates, with an initial unit tangent vector of [1,0] .

For a small increment along the curve δt in the first quadrant, the coordinates are δx = δt . cos δ θ and δy = δt . sin δ θ , where δ θ is the slope of δt . The slope is the tangent δ y δ x , i.e., the tangent vector [ δ x , δ y ] of the transition curve. Thus, arbitrary point P (t) = (x, y) is obtained by integrating along the contour of the transition curve:

P (t) = ( 0 L t cos f ( t ) . d t , 0 L t sin f ( t ) . d t )
where L t = 0t dt = t .
So the length of the curve is the integral of the curve infinitesimal between the origin and P (t) . And the coordinates of P (t) are similar integrals, but are of the projections of the curve infinitesimal on the coordinate axes. The integral yielding the length of the curve is somewhat circular, and potentially confusing, inasmuch as the chosen parameter is the length of the curve from the origin to point P (t) .

With respect to curvature: for a circular arc of length a, radius r, and angle φ subtended by the arc at its centre, the relationship between these characteristics is given by a = r φ . Then, applying this equation to an infinitesimal arc of the transition curve, and re-arranging, yields 1 R = δ θ δ t . Using equivalences from above, and taking the limit, yields the value of the transition curvature at the arbitrary point P (t) as κ f ( t ) = d θ dt = d f ( t ) dt .

The Clothoid
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What has been referred to as the contour function f ( t ) appears to have significant control over the shape and characteristics of the transition curve. This control is not examined, except for two cases.

If f ( t ) = t then κ f ( t ) = d f ( t ) dt = 1 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 ( t ) = t 2 then κ f ( t ) = d f ( t ) dt = 2 t 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.

Boundary Conditions
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It is necessary to specify the curvature at, and the position of, two points on the transition curve in order to identify the curve uniquely. Clearly, the origin end of the curve is well defined; and the other end of the section of transition track under consideration is the practical second choice.

The curvature, κ 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 κ f ( t ) = 2 t t for the clothoid, at the non-origin end of the curve it follows that κ L = 1 R L = 2 L u 2 , where u2 is the normalizing constant. Thus u = 1 2 R L L , and 1 = 2 R L L u 2 . In general at point P (t) , 1 = 2 ( R t u ) ( L t u ) = 2 R N L N , where N means normalized, LN is used to compute the values of the coordinates of P (t) from the integrals, and RN is the radius at the point, computationally unused. Once computed, the values of the coordinates of P (t) must be divided by the normalizing constant to restore the input units values.

Integration
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The integrals for the coordinates of arbitrary point P (t) cannot be evaluated as they stand; this is a simple mathematical result. Instead, integrals of equivalent Taylor Series expansions of trigonometric functions provide a numerical result to the required accuracy. The key issues are addressed straightforwardly on the internet, including a description in Euler Spiral of the Fresnel_integrals that have to be integrated; and the information is not replicated here.
Application
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Since κ f ( t ) = 2 t and κ f ( t ) = d θ dt from above, in general, when normalized, θ clothoid = 0 L 2 t u 2 dt = [ t 2 u 2 ]0 L = L 2 2 R L L . Thus L = 2 R L θ clothoid = R L θ constant-curvature , hence θ clothoid = θ constant-curvature 2 . This is an important result for modular track application because it says that, from an angular point of view, a constant curvature module can be replaced exactly by the combination of a transition and its reverse, an easement. However, the total arc length of the replacement modules is twice that of the constant curvature module.

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.

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last-modification-date:  3 Jan 2016