@CONFERENCE\{IMM2008-05852,
    author       = "H. True and M. Hoffmann",
    title        = "The Dynamics of European two-axle Railway Freight wagons with {UIC} standard Suspension",
    year         = "2008",
    pages        = "225-236",
    booktitle    = "20th Symposium of the International Association for Vehicle System Dynamics, August 13-17 2007",
    volume       = "46",
    series       = "{VSD} Supplement",
    editor       = "J. Karl Hedrick",
    publisher    = "Taylor \& Francis",
    organization = "",
    address      = "",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/5852-full.html",
    abstract     = "The Dynamics of European two-axle Railway Freight
Wagons with {UIC} Standard Suspension

Damping by dry friction has been used in vehicle constructions for centuries, and it is still used in railway vehicles today. The reason is that the dry friction dampers are much cheaper and more robust than the hydraulic dampers are. The drawbacks are that their characteristics are sensitive to weather
conditions, pollution (dirt, oil) and to a high degree to wear. The designers use rules learned by experience for the applications of dry friction dampers in railway vehicles in part because there exist no generally accepted physical laws for their function. In this presentation we shall explain the dynamics of the
European two-axle freight wagons on the basis of sound physical laws and mechanical principles.
The {UIC} Standard Suspension is the internationally approved suspension for two-axle freight wagons in Europe. Each wagon has four of these suspensions. They consist of a leaf spring, which rests on the axle box, which can move horizontally and vertically between the guidances. In the horizontal direction the guidances provide an end stop for the translations and the yaw of the wheelset. In the vertical direction the motion of the wheelset is restricted by the leaf spring with its nonlinear dry friction characteristic and the weight of the car body. The car body is suspended on the leaf spring by a system of links that basically function as double pendulums with a restricted amplitude so the horizontal restoring force is stepwise progressive. Damping is provided by the dry adhesion forces in the cylindrical connections of the elements of the links. We thus deal with a mechanical system that has been used for more than 150 years because it is simple, rugged and cheap in price and maintenance. On the other hand it has eluded theoretical dynamical investigations because of its complexity with dry friction elements and the motion delimiters that create impacts.
We use Piotrowski's theoretical model [3] of the suspension. The characteristics and the parameters of the link system have been measured on a test stand with the aid of Professor Piotrowski in his laboratory in
Politechnika Warszawska, Poland. The German Railways, Minden, Germany, have provided the necessary data for the modelling of the dynamical system of a new Hbbills 311 freight wagon with two axles, which we have chosen as our primary wagon model. Two other types of wagons are also investigated and the dynamics is compared with the dynamics of the Hbbills 311 freight wagon. They are a G69 freight wagon from Pascal [2] and a Kbps wagon that was investigated by J{\"{o}}nsson [1]. The wagons run on an ideal, straight and level track with constant speed. We want first to investigate the dynamics of the wagons in dependence on the speed, which is the bifurcation parameter in the dynamical problem. The equations of motion are formulated in a coordinate system that moves along the track center line with the constant speed of the wagons. The wheel/rail contact geometry is nonlinear. The rails are UIC60 rails and the wheels have S 1002 wheel profiles. Three different cants of the rails are investigated. This nonlinear constraint is coupled with the nonlinear creep-creep force relation in the wheel-rail contact zone. We use Kik's routine {RSGEO} for the contact computations and Shen-Hedrick-Elkins' formula for the creep-creep force relation. Earlier investigations have shown that it delivers an accurate description of the dynamics of the vehicle.
The nonlinear dynamical system is formulated as a system of first order
differential equations with discontinuities. It is solved numerically with appropriate initial conditions for increasing values of the speed. We use a one-step solver with variable steplength and error control. The discontinuities - such as impacts and stick-slip - are formulated as switching boundaries in the phase space. Whenever a trajectory hits a switching boundary an event occurs, and the integration stops. The law of the event is then applied, and the integration continued. Due to the stick-slip in the dry friction model the system does not have a well defined stationary solution at low speed. It is a set valued problem. The results of the dynamic investigation will be presented as bifurcation diagrams and time series. They will be discussed. The wagons have a critical speed above which they oscillate periodically. At higher speeds chaotic motion may develop. The first bifurcation is always a classical Hopf bifurcation. It may seem surprising since the system is non-smooth. It turns out that the cant of the rails have a profound influence on the dynamics of the wagons. Results will be shown for empty as well as for symmetrically and asymmetrically loaded wagons.

References
[1] {P-A}. J{\"{o}}nsson. Modelling and Laboratory Investigations on Freight Wagon Link Suspensions with respect to Vehicle-Track Dynamic Interaction. PhD thesis, Royal Institute of Technology, Stockholm, Sweden, 2004.
[2] {J-P}. Pascal. Oscillations and chaotic behaviour of unstable railway wagons over large distances. Chaos,Solitons and Fractals, 5(9):1725\{1753, 1995.
[3] J. Piotrowski. Model of the {UIC} link suspension for freight wagons. Archive of Applied Mechanics, 73:517\{532, 2003.",
    isbn_issn    = "{ISBN} 978-0-415-48602-6"
}