Sunday, September 06, 2009

Clothoid curves, motorways, and Hermann Tilke

Between the wars...road engineers became obsessed with working out the perfect transition curve - a mathematical method of shifting smoothly between a straight and an arc so that centripetal force builds up gradually, not suddenly like on a fairground ride. The roadbuilders swapped various mathematical formulae until, in 1937, the county surveyor of Devon, Henry Criswell, produced a set of labour-saving tables for plotting beautifully fluid lines that were so user-friendly they knocked rival systems into touch. Criswell became the undispuated king of the 'clothoid curve' - a graceful arc with a slowly increasing curvature that kept motorists permanently on their toes...The M4, designed by computer from the early 1960s onwards, is a gentle series of transition curves from London to south Wales. (Joe Moran, On Roads, p34-35).

Discussion centred on the rate at which centripetal acceleration should be permitted to change. Acceleration had units of feet per second squared so that the rate of change of acceleration was measured in feet per second cubed (ft/sec3). Henry Criswell's tables were devised for 1 ft/sec3, but in later work he also produced tables for 2 ft/sec3, this latter figure was the same as that used in the USA while Australia used 3 ft/sec3.

In the 1940s John Leeming conducted a series of experiments to measure the rate of change of acceleration actually experienced by cars on the road...The speed chosen by drivers led to rates of change of centripetal acceleration up to 10 ft/sec3, which was very much higher than anticipated without any apparent discomfort to the driver or risk to safety. [Leeming's work] seems not to have been put to any widespread use despite the implication that transitions could be much shorter.
(John Porter, The Motorway Achievement: Frontiers of knowledge and practice, p129).

This description of clothoid curves immediately reminds one of Hermann Tilke's Formula 1 circuit design ethos. In particular, turns 1 and 2 at Shanghai, pictured here, seem to have been lifted from the higher curvature parts of a clothoid spiral. Which almost tells you everything you need to know about modern F1 circuit design: the corners are drawn from the same palette of curves used by motorway architects.


Grinnyguy said...

My physics teacher used to talk about sexy graphs. Now that is a sexy graph! Really interesting research with a sensible use for it afterwards - the best kind of research

joam said...

Hi there,

I am a civil engineering student, at my transport module i was asked to solve a Q to find the radius at which 2 clothoids of different radial accelarations meet ( in a horizontal allignment), i have been wandering the streets trying to think of something, may you suggest a helpful tip ?

Gordon McCabe said...

I'm afraid that's beyond me at the moment Joam!