Engineering
Railway bogie
Run a wheelset with coned treads along a test track and watch the path it takes drawn on the ballast, thirty times enlarged. A free wheelset rolls Klingel’s sinusoid, 2 pi sqrt(b r0/lambda) long; tied to its bogie by lateral and yaw springs it is stable up to the critical speed the Routh-Hurwitz test gives, and a kick dies away - go ten per cent faster and the same kick grows into hunting from flange to flange. Wear the treads to a steeper cone and the critical speed drops, stiffen the yaw spring and it rises; put the track into a curve and the wheelset settles at the offset where creep and suspension balance.
What happens
One wheelset of 1250 kg with 1:6.7 coned treads (lambda 0.15) and wheel radius 0.46 m, tied to its bogie by a lateral spring of 1 MN/m and a yaw spring of 2 MN·m/rad. Kalker’s linear creep coefficients of 8 MN give the longitudinal and lateral forces, saturating smoothly at the adhesion limit; the flanges stop it 7.5 mm off centre with a stiff spring. A free wheelset rolls Klingel’s sinusoid 2 pi sqrt(b r0/lambda); the quartic of the suspended wheelset gives the critical speed by the Routh-Hurwitz test. In a curve the wheelset settles where creep and suspension balance. RK4 at 0.5 ms; the path is drawn thirty times enlarged.
Try this
Kick the wheelset below the critical speed and watch it settle; go ten per cent faster and see it hunt; wear the treads to a steeper cone; stiffen the yaw spring; put the track into a curve.
The model and its limits
A single wheelset with linear creep and a smooth saturation, a constant conicity, a spring flange, no bogie frame dynamics or second wheelset. Critical speeds are those of this idealized wheelset.
Equations used in the module
Equations m·a_y = −2f22·(v_y/V − ψ) − k_y·y − F_flange I·α_ψ = −2b·f11·(b·ω_ψ/V + λ·y/r0 − b/R_c) − k_ψ·ψ λ_K = 2π·√(b·r0/λ) a3·a2·a1 − a3²·a0 − a1²·a4 = 0 → V_crit
