A codimension two bifurcation in a railway bogie system

Abstract

In this paper, a comprehensive analysis is presented to investigate a codimension two bifurcation that exists in a nonlinear railway bogie dynamic system combining theoretical analysis with numerical investigation. By using the running velocity V and the primary longitudinal stiffness $K_{1x}$ as bifurcation parameters the first and second Lyapunov coefficients are calculated to determine which kind of Hopf bifurcation can happen and how the system states change with the variance of the bifurcation parameters. It is found that multiple solution branches both stable and unstable coexist in a range of the bifurcation parameters which can lead to jumps in the lateral oscillation amplitude of the railway bogie system. Furthermore, reduce the values of the bifurcation parameters gradually. Firstly, the supercritical Hopf bifurcation turns into a subcritical one with multiple limit cycles both stable and unstable near the Hopf bifurcation point. With a further reduction in the bifurcation parameters two saddle-node bifurcation points emerge, resulting in the loss of the stable limit cycle between these two bifurcation points.

Appendix: Description and values of the railway bogie system parameters

Parameter	Description	Value
$M_{t}$	Mass of the bogie frame	2056 kg
$J_{tz}$	Yaw moment of inertia of the bogie frame	$3800~\mathrm{kg\,{m}}^2$
$M_{w}$	Mass of the wheelset	1627 kg
$J_{wz}$	Yaw moment of inertia of the wheelset	$830~\mathrm{kg\,{m}}^2$
$K_{1x}$	Primary longitudinal stiffness (per axle box)	–
$K_{1y}$	Primary lateral stiffness (per axle box),	$7.0~\mathrm{MN\,{m}}^{-1}$
$C_{1x}$	Primary longitudinal damper (per axle box)	$0.0~\mathrm{MN\,{s}\,{m}}^{-1}$
$C_{1y}$	Primary lateral damper (per axle box),	$0.0~\mathrm{MN\,{s}\,{m}}^{-1}$
$K_{2x}$	Secondary longitudinal stiffness (per side of bogie)	$0.133~\mathrm{MN\,{m}}^{-1}$
$K_{2y}$	Primary lateral stiffness (per side of bogie),	$0.133~\mathrm{MN\,{m}}^{-1}$
$C_{2x}$	Primary longitudinal damper (per side of bogie)	$0.0~\mathrm{MN\,{s}\,{m}}^{-1}$
$C_{2y}$	Primary lateral damper (per side of bogie),	$0.015~\mathrm{MN\,{s}\,{m}}^{-1}$
$r_0$	Centered wheel rolling radius	0.46 m
b	Half of the rolling cycle gauge	0.7465 m
$b_1$	Half of the swing arm	1.02 m
$b_2$	Half distance of the secondary springs	0.95 m
$b_3$	Half distance of the secondary dampers	1.275 m
l	Half of the axle distance	1.25 m
v	Running speed of the bogie	–
$f_{11}$	Longitudinal creep coefficient	12 MN
$f_{22}$	Lateral creep coefficient	12 MN
W	Axle load	103.936 kN
$\lambda$	Conicity of the wheel when $y_{wi}=0$	0.2
$\delta _1$	Nonlinear coefficient of wheel–rail contact force	$-1.6\times 10^{11}~\mathrm{N\,{m^{-3}}}$
$\delta _2$	Nonlinear coefficient of wheel–rail contact force	$1.6\times 10^{15}~\mathrm{N\,{m^{-5}}}$