Wheelset Hunting Rig (3D)

Why a train sways: the conical wheel

Railway wheels are not cylinders. Their treads are slightly conical with conicity lambda (1:20 is 0.05, a worn profile can reach 0.3 or more). When the axle shifts sideways by y, the two wheels roll on different radii, r_right - r_left = 2 lambda y. The wheel on the larger radius covers more track per revolution, so the axle swings back towards the centre line, overshoots, and sways: hunting.

Klingel's kinematic wavelength

If the wheels roll without any slip, the axle follows y'' + (lambda V^2 / (b r0)) y = 0 with b the half gauge between contact points (0.7175 m) and r0 = 0.46 m the wheel radius. The path is a sine of wavelength

L_K = 2 pi sqrt(r0 b / lambda)

independent of speed (about 8 m for lambda = 0.2). The frequency then rises with speed, f = V / L_K.

Creep forces and the Wickens model

Real wheels do slip a little (creep), and the resulting forces are what damp or feed the motion. For small creepages the force is proportional to the slip, with creep coefficient f (about 10 MN). With a single wheelset restrained by lateral stiffness k_y and yaw stiffness k_psi (the primary suspension) and optional dampers c_y, c_psi:

m y'' + (2f/V + c_y) y' - 2f psi + k_y y = 0
I psi'' + (2 f b^2/V + c_psi) psi' + (2 f lambda b / r0) y + k_psi psi = 0

Substituting exp(s t) gives a fourth-order characteristic polynomial. Below the critical speed all four roots lie in the left half-plane and a disturbance dies out; above it the least damped pair crosses into the right half-plane and the oscillation grows. The critical speed falls roughly as 1/sqrt(lambda) and rises with yaw stiffness and with a yaw damper, which is why high-speed bogies have stiff yaw links and yaw dampers.

Flange limit cycle

In the time domain the creep forces saturate at the friction limit (mu N, with mu = 0.35 and 110 kN axle load) and the flange stops the motion at the clearance delta (a stiff contact spring of 20 MN/m). Above the critical speed the amplitude therefore does not grow forever: it settles into a limit cycle that bangs against the flange, producing the noise, wear and derailment risk that limits real train speed.

Numerical checks on this page

(1) The quartic of the full model with infinitely stiff creep and no restraint is solved numerically and its wavelength compared with Klingel's formula. (2) The critical speed from the root crossing is compared with the sign change of the third Routh-Hurwitz determinant, an independent analytic criterion. (3) A 4th-order Runge-Kutta time integration of a tiny disturbance gives a growth rate compared with the real part of the dominant root.

Honest simplifications

One wheelset with linear (Kalker-type) creep coefficients identical in both directions and a constant effective conicity; a real bogie has two coupled wheelsets, a frame and a vehicle body, nonlinear wheel and rail profiles (conicity changes with amplitude), spin creep, gravitational stiffness, track flexibility and irregularities. The numbers (critical speed in km/h) are plausible orders of magnitude for a teaching model, not data for a particular vehicle. The lateral shift is exaggerated visually (slider) because real displacements are millimetres. The tread slope on the 3D wheel is drawn at 1:10 for visibility whatever conicity you choose.

What to try

• Load Fast, just below critical and press Kick the wheelset: the sway decays. Then load Above critical: flange hunting: it grows until the flange stops it.
• Press Speed ramp and watch graph 2: the damping ratio of the least damped mode falls through zero at the critical speed.
• Raise conicity to 0.4 (worn wheels) and see the critical speed drop on graph 4; the wavelength on the track plan gets shorter.
• Add the yaw damper (c_psi 120 kN m s/rad): the critical speed jumps well above 400 km/h.
• Reduce the creep coefficient (wet rail): with less friction the damping changes and the critical speed moves.