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Railway physics, with two cups

How a train steers without a steering wheel

A train wheel is shaped like a cone, and the shape does the steering. Push the axle sideways and watch it find its own way back, in a wave about ten metres long.

Length of one weave 9.5 m

Sideways movement is drawn about 400 times larger than distances along the track, so that it shows. With the wheelset centred, the nominal play between flange and rail is about 5 mm on each side.

The difference in size is drawn 150 times too large.

Both wheels are fixed to one axle, so they turn together. Shift the axle a few millimetres to the right and the right wheel ends up running on a wider part of its cone: in each turn it covers more ground than the left wheel. The axle swings back to the left, overshoots the middle, and swings again. In this ideal picture the length of that wave is about ten metres and depends on the wheel and the rail, not on how fast the train goes. The German engineer Klingel worked out the formula in 1883.

Can two cups steer like a train?

Take two cups with a plain top edge (the published activity uses plastic or foam ones) and tape them together, rim to rim. Stand two long rulers, metre sticks or yardsticks on their long edges, parallel, rest one end of each on a book and tape them in place to make a ramp, then let the cups roll down. An activity from Science Buddies, published in Scientific American in 2019, says what to expect: cups taped rim to rim stay on the track even when you start them off-centre and steer themselves back towards the middle, while cups taped base to base stay on only if they start perfectly centred, which is almost impossible.

How the cups sit on the rulers. Two cups taped rim to rim, seen from the front, resting on two rulers standing on their long edges. Each cup touches a ruler halfway up its side.rimbasebasegap between the edges = height of one cupruler top edgeruler top edge
Two cups taped rim to rim, seen from the front, resting on two rulers standing on their long edges. Each cup touches a ruler halfway up its side.

Measure your cups, and Klingel's formula tells you how long a weave to look for.

Predicted weave: about 69 cm. A metre stick covers about 1.4 weaves.

With a gap equal to the cup height, each cup touches a ruler halfway up its side. A narrower gap or smaller cups give a shorter weave.

Rim to rim: the cups weave from side to side and stay on the rulers.

The formula assumes stiff cones rolling without slipping. Cups flex and slip a little, and a ruler is not a rail, so the weave you measure can differ from the prediction by a good margin.

How long is the weave of a real wheel?

The formula that fits a cup fits a wheel. A new passenger wheel is 920 millimetres across, so its radius is 0.46 metres, and the points where the two wheels of an axle touch the rails are about 1.5 metres apart. The last number, the taper of the cone, is less certain than it looks.

Weave length of a 920 mm wheel, by taper
TaperWeaveWhere the taper comes from
0.0516.5 mThe textbook figure for the cone: 1 in 20
0.1011.7 mLower end of the typical range, for smooth running at a 3 mm sideways shift
0.159.5 mA new wheel on a standard rail, from a published calculation
0.257.4 mDesign limit for European track at 60 to 200 km/h, checked with new wheels

The 1 in 20 is a nominal value. Real wheel treads are curved, and the rails lean inwards (1 in 40 in Germany, Austria and Switzerland, 1 in 20 in Britain, France and Italy), so engineers measure an equivalent conicity instead. For a new S1002 wheel on a UIC 60 rail leaning 1 in 40, a published calculation gives about 0.15 for small sideways shifts; on a rail leaning 1 in 20 the same wheel gives about 0.05. Countries with rails leaning 1 in 20 use wheel profiles shaped for them (Britain's P8 profile gives about 0.13 on such rails in the same study). Either way, the weave of a real wheel is nearer ten metres than seventeen.

Klingel, an engineer in Karlsruhe, printed the formula in 1883 in a German railway journal, together with a table of weave lengths. The table does not state the rail spacing, and a spacing of 1.5 metres, centre to centre, reproduces all eight of his figures. For a taper of 1 in 20 and a wheel radius of 450 mm the formula gives 16.32 metres, as his table does.

All eight rows of Klingel's table, rebuilt
TaperWheel radius, mmWeave, m
1 in 2045016.32
1 in 2050017.21
1 in 2055018.05
1 in 2060018.85
1 in 1645014.60
1 in 1650015.39
1 in 1655016.14
1 in 1660016.86

Every row matches the printed table to the hundredth of a metre.

How does it take a bend when both wheels turn together?

On a bend the outer rail is longer than the inner one, yet a solid axle turns both wheels at the same rate. A car solves this with a differential. A train has none. The axle slides outwards until the outer wheel runs on a wider part of its cone, so it covers more ground per turn and the wheelset follows the curve. All of it happens inside the play between flange and rail, which is about 10 millimetres in total.

Axle shift needed: 2.3 mm of the 5 mm available

Shift = wheel radius × half the contact spacing ÷ (taper × radius of the bend), with a taper of 0.15. This is the shift of a straight cone; real treads are curved and steepen towards the flange, so they cope with somewhat tighter bends. In sharp bends the flange does touch the rail, and some locomotives and multiple units carry lubricators that grease it for that reason.

How many times a second do the wheels under your seat weave?

Pick a line, or type a speed. In Klingel's picture the weave length is the same at any speed, so speed decides how often the wheels meet a wave.

2.9 weaves per second, for a free wheelset

2.4 to 3.8 for conicities of 0.10 to 0.25

At 100 km/h the wheels cover 28 metres every second.

These are the figures for a wheelset free to move. In a real bogie the frame couples two axles and stretches the wave. On a Chinese train running at 350 km/h, researchers measured a bogie weaving 8.2 times a second, on worn wheels whose equivalent conicity was 0.55 at a 3 mm shift. For that conicity Klingel's formula gives about 20 weaves a second for a free wheelset and about 10 for a rigid bogie. In that train, worn contact between wheels and rails made the bogie weave at a frequency close to one of the carriage body's own vibration modes, and passengers felt strong shaking. A study of metro trains likewise ties ride quality to how close the weave comes to the carriage's own sway frequency.

Why is the cone also a speed limit?

Real wheels slip a very little as they roll, and the slip creates forces where wheel meets rail. At low speed those forces calm the weave down. Above a critical speed they feed it, and the weave grows until the flanges hit the rails. Engineers call this hunting. A. H. Wickens, who analysed it in the 1960s, described it as a self-excited oscillation caused by the conicity of the wheels together with the creep forces.

At this speed the weave dies out. In this model the weave starts to grow at 200 km/h.

More taper makes bends easier and lowers the speed at which the weave starts to grow. Every wheel profile is a compromise.

The model is a textbook one: a single wheelset on springs, with parameters from a published paper. At 0.05, 0.15 and 0.25 it starts to grow at 350, 200 and 155 km/h. A real train also has a bogie and a body, and its limit can lie far from these figures. A published model of a Spanish high-speed bogie crosses its limit at 427 km/h. In July 2014, near Brockville, Ontario, one truck of an empty Canadian freight car likely hunted excessively at about 60 mph (97 km/h), with worn parts contributing, and 26 cars derailed.

The engineer who found the wave in the bent rails of a derailed train

On 1 July 1947 a passenger train derailed and overturned on the Sanyō line in western Japan, between Hikari and Kudamatsu, and 15 people died. Among the investigators was Tadashi Matsudaira, a former aircraft engineer. He had graduated in naval architecture in 1934, joined the Japanese Navy's aircraft arsenal and investigated why Zero fighters broke up in the air. For the prototype, the finding was elevator flutter, a vibration that feeds itself, set off by a mass balance that earlier shocks had broken. After the war he had moved to the Railway Technical Research Institute.

At the site, the rails just before the point of derailment were still in one piece, but over about 50 metres they curved left and right in a sine wave. His senior colleagues showed no interest: the bend, they said, had been there before, and the accident had only made it stand out. Matsudaira thought the train's own sideways swing had marked the rail and then thrown it off. As Matsudaira recalled it, railways knew a left-right motion called hunting, but the usual explanation blamed faults in the track, and the idea of a vibration that sustains itself was hardly known.

He built a model: a small vehicle standing on rotating wheels that played the part of rails. The model rails were parallel, and at speed the model began to shake violently on its own. At some speed, Matsudaira described, the body suddenly swings strongly from side to side; faster, that swing disappears; faster still, the body hardly moves but the wheels shake violently from side to side. He showed the experiment to many people and, he said, from then on hunting was correctly understood.

At the Institute he led the group studying how high-speed vehicles move, and took part in designing the bogie of what became the 0 Series Shinkansen. A later report says the hunting of the test trains was so violent that the track was often deformed, and that former fighter designers applied their flutter analysis to bring it under control. The test trains reached 200 km/h in October 1962 and 256 km/h on 30 March 1963. The Tokaido Shinkansen, from Tokyo to Osaka, opened on 1 October 1964. Matsudaira died on 4 August 2000, aged 90.

What to look at on the next platform

Step back from the edge and look at the top of the nearest rail. The bright strip of polished steel is where the wheels run. Each of those wheels is shaped like a cone.

The cups need five minutes and no skill. If your weave turns out longer or shorter than predicted, that is the interesting part.

Sources and method

Standard or published: the gauge (1,435 mm), the 1.5 m between contact points, the 920 mm wheel, Klingel's formula and table, the conicity limit of 0.25, and the play between flange and rail, which follows from standard wheelset dimensions (1,435 − (1,360 + 2 × 32.5) = 10 mm in total).

Derived here: every weave length, the weaves per second and per trip, the curve limit, the cup prediction and the speeds in the model. The equivalent conicity of 0.15 for a new wheel is read off a published plot, give or take 0.02.

Not measured: the cup prediction. The speed-limit model is a textbook one-wheelset model with literature parameters; its figures show how the limit moves with taper, not where a given train hunts.

Rail distances: Bucharest–Constanța 224.8 km, London–Edinburgh 632 km, Tokyo–Osaka 515.4 km. Line speeds are the maximum allowed or reached in service.