Basic Model of a Wheelset, Degrees of Freedom

For more than 150 years, the wheel – rail system has provided a relatively safe system of transport. This safety level is so high that the mechanism is generally neglected and considered as a simple slider by most people.
However, the engineer’s point of view can be different, especially when taking into account responsibilities in a railway network. The wheel – rail contact is actually a complex and imperfect link. Firstly, it is a place of highly concentrated stresses. The conical wheel shape makes the wheelset a mechanical amplifier, limited by the transverse play, with partially sliding surfaces. The contact surfaces are similar to those in a roller bearing but without protection against dust, rain, sand, or even ballast stones.

If the track is considered to be rigid, then the railway wheelset has two main degrees of freedom:

• The lateral displacement, or shift, y
• The yaw angle, a

When the behaviour of a wheelset is unstable, the dynamic combination of these two degrees of
freedom is called “hunting.”
The lateral displacement and the yaw angle must be considered as two small displacements relative to the track. The play will be the limit of the lateral displacement between the two flange contacts. It is generally approximately +-8 mm.
The other degrees of freedom are constrained: the displacement along Ox and the axle rotation speed v around Oy are determined by the longitudinal speed Vx and the rolling radius of the wheel r0 with: Vx ¼ vr0: The wheelset centre of gravity height z and the roll angle around Ox are linked to the rails when there is contact on both rails.
The railway wheelset is basically described by two conical, nearly cylindrical wheels (Figure 1 and Figure 2), linked together with a rigid axle.

Figure 1 Wheelset degrees of freedom.

Figure 2 Rail, wheel and contact frames.

Each wheel is equipped with a flange, the role of which is to prevent derailment. In a straight line the flanges are not in contact, but the rigid link between the two wheels suggest that the railway wheelset is designed to go straight ahead, and will go to flange contact only in curves. This is the railway dicone or wheelset.
The interface between the wheel and the rail is a small horizontal contact patch. The contact pressure on this small surface is closer to a stress concentration than in the rest of the bodies. The centre of this surface is also the application point of tangential forces (traction and braking Fx, guiding or parasite forces Fy, see Figure 1). The knowledge of these forces is necessary to determine the general wheelset equilibrium and its dynamic behaviour.
In order to determine this behaviour and these forces, the first thing to do is to determine some contact parameters: the contact surface, the pressure and the tangential forces. This determination is generally separated into two steps:

1. The normal problem (Hertz theory)
2. The tangential problem (Kalker’s theory)