One of the beauties of measuring angles in radians (of which there are 2
“The circumference is the radius times the radians.” This can be used to great effect in calculating the superelevation (crossfall or banking) that needs to be applied to keep a train traveller’s soup level, or a motorcycle rider ‘upright’.
A friend lent me the BR bible on such matters, the “CIVIL ENGINEERING HANDBOOK No. 3. RULES FOR SPEED OF TRAINS ON CURVES IN RELATION TO RADIUS, CANT, AND LENGTH OF TRANSITION”, July 1962. [1]
This has detailed graphs and charts as a ready rekoner on how much higher the outer rail should be laid depending on the expected speed of the train. The text is of course in the English vernacular units of inches, miles per hour and chains. Also notable is the proper use of the Oxford comma in the title.
The basic mathematics is thus:
Considering a train travelling along a circular track. In a small distance it will change direction by a small angle. If at midday it was travelling north then, some short time later it will be travelling east or west to some extent.
If it’s speed is v, and the radius of the track is R, then it will be turning at v/R radians per second. In 00 scale, if the train is running at 24 inches per second on a curve of 18″ radius, then it will be turning at 1.1/3 radians per second.
In full size, v is in mph and R is in chains (there are 80 chains in a mile), so there is also a constant to take care of, namely 80 (chains in a mile) / 3600 (seconds in an hour). A train at 120mph on a 1600 chain curve therefore turns at (120 × 80) / (3600 × 1600) or 1/30 × 8/160 or 1/3 × 1/100 × 1/2, or 1/600 of a radian per second.
Note: this is also the yaw rate of the train, that is the speed at which a compass turns when viewed by a passenger on board. Although that would probably be noted in degrees.
The turning of the train imparts a new speed of the train in the direction of the new heading. Where previously the northbound train had no speed east-westwards, it now has a speed of v times its new heading (in radians). This change in velocity means it can be considered to be accelerating at a rate of its speed times its angular velocity, this in a direction perpendicular to its motion, or centripetally – towards the centre of the circle. The passengers are also being dragged centripetally, to stop them continuing north, and describe their outwards tendency as a centrifugal force.
The centripetal acceleration has to be generated by a lateral force. A train ‘pottering’ round a 12c radius viaduct [2]
at 30 mph would have an angular velocity of (30×80)/(3600 ×12) or 1/18 of a radian per second and a lateral acceleration of this times 30 x (80 x 66/3600) — the speed expressed in feet per second– or 44/18 feet per second per second (‘/s/s). [2]
On the basis that a 1lb force (lbf) accelerates a 1lb mass by 32’/s/s, if the train weighed 100tons it would take 100 × 20 × 112 × (44/18) / 32 or … 224,000 x 11/144 pounds of lateral thrust [3].
This is also the same force that would be needed to pull it up an 11 in 144 gradient, so if the rails are set with this amount of crossfall (on 4′ 8.1/2″ track that would be one rail about 4″ higher than the other) then the train’s tendency to fall to one side is exactly balanced by its tendency to swerve outwards on the curve.
The chart from the BR handbook gives 4.1/2″
Note that in all this the mass/ weight of the train has the same effect in both arguments, a train of twice the weight would require twice the force, but the slopes would be the same. So the weight becomes irrelevant in the debate over superelevation. The crossfall for neutral steering only therefore depends on the speed and the radius of the curve.
To experimentally validate this, a circle of Hornby 2nd radius track was set up and a plate made with three spirit level vials set at nominally 0, 5% and 10% inclination.
These were calibrated on a surface table, set level with a scanning laser. With the plate 1.500″ wide, the height difference to align the bubbles was measured with a digital height gauge. The vials actually measured 5.65% and 11% crossfall.
At these crossfalls, the train would have to travel at about 20 and 27.1/2 “/s. This would make a lap time of 5.7 and 4.1s for each of the inclined vials to appear level.
[1] British Transport Commission, British Railways, CIVIL ENGINEERING HANDBOOK No. 3, July 1962
links to pdfs of part 1 ; part 2 ; part 3
[2] Railscot entry on Glenfinnan Viaduct
[3] There are 66 feet in a chain which is 1/80 of a mile.
[4] There are 20 hundredweight (cwt) of 112lb in a ton.
philbarber2 