Hans True

Bifurcations and chaos in a model of a rolling railway wheelset

In this paper we present the results of a numerical investigation of the dynamics of a model of a suspended railway wheelset in the speed range between 0 and 180 km h-1. The wheel rolls on a straight and horizontal track unaffected by external torques. A nonlinear relation between the creepage and the creep forces in the ideal wheel rail contact point is used. The effect of flange contact is modelled by a very stiff spring with a dead band. The suspension elements have linear characteristics, and the wheel profile is assumed to be conical. All other parameters than the speed are kept constant. Both symmetric and asymmetric oscillations and chaotic motion are found. The results are presented as bifurcation diagrams, time series and Poincaré section plots. We apply bifurcation and path following routines to obtain the results. In the last chapter we examine one of the chaotic regions with the help of symbolic dynamics.

PARAMETER STUDY OF HUNTING AND CHAOS IN RAILWAY VEHICLE DYNAMICS.

SUMMARY We investigate the dynamics of Cooperriders bogie model with realistic wheel and rail profiles. The results are presented mainly as bifurcation diagrams We find speed ranges with asymptotically stable forward motion along the track centre line, coexisting attractors, symmetric and asymmetric oscillations and chaos. In contrast to earlier investigations, we also find chaotic transients, which lead to derailment, before the motion might have settled down to stable motion The influence of a change in the coefficient of adhesion, the wheel base, the gauge and the cant of the rails on the bifurcation points is discussed.

Dynamics of a model of a railway wheelset

In this paper we continue a numerical study of the dynamical behavior of a model of a suspended railway wheelset. We investigate the effect of speed and suspension and flange stiffnesses on the dynamics. Numerical bifurcation analysis is applied and one- and two-dimensional bifurcation diagrams are constructed. The onset of chaos as a function of speed, spring stiffness, and flange forces is investigated through the calculation of Lyapunov exponents with adiabatically varying parameters. The different transitions to chaos in the system are discussed and analyzed using symbolic dynamics. Finally, we discuss the change in orbit structure as stochastic perturbations are taken into account.

RAILWAY VEHICLE CHAOS AND ASYMMETRIC HUNTING

SUMMARY In this paper we present the results of a refined investigation of the dynamical behaviour of Cooperriders complex bogie. The earlier results were presented in [4] and [7]. It was discovered, that one of the solution branches in [4] and [5] was one of an asymmetric, periodic oscillation - albeit with a very small offset, but it indicates, that the asymmetric oscillation is the generic mode at speeds much lower than has hitherto been found. The bifurcation diagram has been completed, a new type of bifurcation discovered and the other asymmetric branch determined. Furthermore we discovered chaotic motion of the bogie at much lower speeds than reported in [5] and [8][, and we present the result here. Finally we present a new solution branch, which represents an unstable, symmetric oscillation. It has the interesting property, that it turns stable in a small speed range for very high speeds. It has a smaller amplitude than the coexisting chaos. Such behaviour is not uncommon in dynamical systems, see...

On the dynamics of the three-piece-freight truck

Although the three-piece-freight truck is a simple design its mathematical model is very complicated. The model is definitely a nonlinear dynamical system, where the nonlinearities arise from the nonlinear kinematic and dynamical contact relations between wheels and rails, the suspensions and the nonlinear dry friction damping with hysteresis and stick-slip action. The bolster moves both vertically and laterally relative to the truck frames, so the friction forces on the contact surfaces of the wedges must be treated as two-dimensional vectors, and the same holds for the dry friction on the surfaces of the adapters. Due to the clearances between the car body and the side supports on the bolster, the side supports must be modelled as nonlinear dead-band springs. The stick-slip action and the play between elements of the truck makes the dynamical model a structure varying system. We present the dynamical system that models the dynamics of the moving wagon and show the result of numerical dynamical investigations such as the calculation of the critical speed, and the dynamics of the wagon on an irregular track and compare them with test results and simulation results using NUCARS.

PERIODIC, BIPERIODIC AND CHAOTIC DYNAMICAL BEHAVIOUR OF RAILWAY VEHICLES

SUMMARY We examine first a simplified model of a railway car on bogies running with constant speed - we describe stationary, periodic, and biperiodic motion. Next we examine the model of Cooperriders complex bogie running with constant speed - we briefly describe the chaotic motion that develops at high speed. In both cases the dynamical equations governing the dynamical motion of the model consist of a set of 14 first order nonlinear ordinary differential equations. The dynamical motions are determined numerically. Several phenomena that are known from nonlinear dynamics are found in our models and they are discussed in the paper.

The dynamics of European two-axle railway freight wagons with UIC standard suspension

The dynamics of two-axle railway freight wagons with the UIC standard suspension is investigated theoretically and the dynamic behaviour is explained. Fully nonlinear models are considered. The hysteresis from dry friction and the effect of impacts between elements of the suspension are included. Different wheel–rail geometries are investigated and the results commented. Bifurcation diagrams are used to describe the eigen-dynamics of the wagons.

On a New Route to Chaos in Railway Dynamics

Cooperriders mathematical model of a railway bogie running on a straight track has been thoroughly investigated due to its interesting nonlinear dynamics (see True [1] for a survey). In this article a detailed numerical investigation is made of the dynamics in a speed range, where many solutions exist, but only a couple of which are stable. One of them is a chaotic attractor.Cooperriders bogie model is described in Section 2, and in Section 3 we explain the method of numerical investigation. In Section 4 the results are shown. The main result is that the chaotic attractor is created through a period-doubling cascade of the secondary period in an asymptotically stable quasiperiodic oscillation at decreasing speed. Several quasiperiodic windows were found in the chaotic motion.This route to chaos was first described by Franceschini [9], who discovered it in a seven-mode truncation of the plane incompressible Navier–Stokes equations. The problem investigated by Franceschini is a smooth dynamical system in contrast to the dynamics of the Cooperrider truck model. The forcing in the Cooperrider model includes a component, which has the form of a very stiff linear spring with a dead band simulating an elastic impact. The dynamics of the Cooperrider truck is therefore “non-smooth”.The quasiperiodic oscillation is created in a supercritical Neimark bifurcation at higher speeds from an asymmetric unstable periodic oscillation, which gains stability in the bifurcation. The bifurcating quasiperiodic solution is initially unstable, but it gains stability in a saddle-node bifurcation when the branch turns back toward lower speeds.The chaotic attractor disappears abruptly in what is conjectured to be a blue sky catastrophe, when the speed decreases further.

Multiple attractors and critical parameters and how to find them numerically: the right, the wrong and the gambling way

In recent years, several authors have proposed ‘easier numerical methods’ to find the critical speed in railway dynamical problems. Actually, the methods do function in some cases, but in most cases it is really a gamble. In this article, the methods are discussed and the pros and contras are commented upon. I also address the questions when a linearisation is allowed and the curious fact that the hunting motion is more robust than the ideal stationary-state motion on the track. Concepts such as ‘multiple attractors’, ‘subcritical and supercritical bifurcations’, ‘permitted linearisation’, ‘the danger of running at supercritical speeds’ and ‘chaotic motion’ are addressed.

Bifurcation of the stationary Ekman flow into a stable periodic flow

The two cases of stationary Ekman boundary layer flow of an incompressible fluid near i) a plane boundary and ii) a free surface with constant shear are considered. It is proven that a stable secondary flow in the form of traveling waves bifurcates from the stationary flow at a certain Reynolds number, and that the stationary flow is unstable above this number. The values of the critical Reynolds number and of the numbers that characterize the traveling wave are computed and compared with experimental values.