The train wheels

Many boys wanted to go into the train dri­ver cab. In fact, going in a straight tun­nel, which is sim­ply a cylin­der, it is not so excit­ing. But some­thing dif­fer­ent are the bends.

But in which man­ner the train wag­ons cor­ner on the rails? In fact, the right and left wheels are rigidly attached together. More­over, in a curve the radius of cur­va­ture of the outer rail is greater than the radius of the inner one, so the length of the arc of a cir­cle which is cov­ered by the out­side wheel is greater than the length of the arc cov­ered by the wheel inside. And it is nec­es­sary that none of the two wheels skid on the rail.

Con­sider a sim­pli­fied model which nev­er­the­less explains well what essen­tially hap­pens. The wheels in this model are two iden­ti­cal trun­cated cones and are rigidly fixed to the axis. The pro­file of the rail sur­face is an arc of a cir­cle. If the rails were embed­ded ver­ti­cally, they would suf­fer an extra boost, since the load of the wheel affects on the rail per­pen­dic­u­larly to its sur­face. There­fore, the rails are embed­ded a lit­tle inclined inwards, so that the axis of their pro­file is per­pen­dic­u­lar to the sur­face of the con­i­cal wheels.

Let us fol­low a pair of wheels while pass­ing on a bend. It hap­pens that they shift with respect to the track!

Observe how a curve is made. After a straight stretch begins a curved one with a vari­able radius of cur­va­ture, then there is a piece with con­stant radius, i.e., an arc of a cir­cle. To ensure that pas­sen­gers do not suf­fer jolts from the sides of the wagon while pass­ing between two dif­fer­ent stretches, a strong con­di­tion must be sat­is­fied: the sec­ond deriv­a­tive of the tra­jec­tory must be con­tin­u­ous at the points of tran­si­tion between two con­sec­u­tive stretches.

The tan­gency between the conic wheels and the rail sur­face is made by points. Mov­ing on a bend, the point of tan­gency moves on the cone. Let us draw the cir­cles made by all points of tan­gency. When the axis is located exactly in the mid­dle, i.e., when the wagon is mov­ing in a straight stretch of track, these cir­cles on the two wheels attached to the same axis are equal. But when the axis is shifted, in a curved stretch, the cir­cle on the inner wheel is smaller than the cir­cle on the outer wheel. So we can con­sider that the wheels have vari­able radii. And since the radius of the inner wheel is smaller than the radius of the outer wheel, when the wheels are mov­ing on the rails with­out skid­ding, the inner wheel makes a shorter ride than the outer wheel.

But why in a bend the pair of wheels is shifted over the rails? It turns out that physics is not involved. It’s just geom­e­try, and also a very beau­ti­ful geom­e­try! When the tracks bend, they cause a shift of the pair of wheels exactly as much as nec­es­sary so that the inner and the outer wheels roll with­out skid­ding.

A good model must take into account the most impor­tant prop­er­ties of the objects under con­sid­er­a­tion and to neglect the details. Such is the model we have con­sid­ered, but in real­ity every­thing is more com­plex. Accord­ing to the rail­way com­pany, the sur­face of the wheel is com­posed of two trun­cated cones with slightly dif­fer­ent open­ings angles. There is more­over a ledge, i.e., the wheel has an edge that pre­vents it from leav­ing the rails. The pro­file of the pro­jec­tion is made of dif­fer­ent arcs of cir­cles. The pro­file of the rail, which in the model con­sisted of a cir­cu­lar arc, is in fact formed by the union of five dif­fer­ent arcs of cir­cles.

How to deal with a bend by a pair of wheels of the train is sub­stan­tially dif­fer­ent from how to deal by the wheels of a car, but in real­ity, in either case plays a role that won­der­ful sci­ence which is the geom­e­try.

Some curios­ity:

  • A pair of train wheels holds about a ton. As for dri­ving wheels, then a more engine is fixed to the axis, which makes reach­ing the load of a ton and a half. The wheel diam­e­ter is about one meter.
  • In Rus­sia, for strate­gic rea­sons the dis­tances between the rails of the train and of the sub­way lines are the same. The track gauge, i.e., the dis­tance between the inner edges of the rails is 1520 mil­lime­ters.
  • A bend is con­sid­ered to be sharp if its cur­va­ture radius is less than 350 meters.. In such curves the gauge can exceed a bit the stan­dard width, but at most by 3 cm.
  • The total length of rail­ways in Rus­sia is about 87 thou­sand kilo­me­tres. The length of sub­way lines in Moscow (in 2011) is about 300 km
  • Some points of each wheel moves in the oppo­site direc­tion with respect to the train. Can you tell which?