Nonlinear dynamic behaviour of a railway wheelset

Nonlinear dynamic systems are well known to contain certain characteristic additional phenomena compared with linear systems. One example is the fact that nonlinear systems can have multiple stable solutions for one set of parameters. In that case, which one of the multiple stable solutions will be realised will depend on the initial conditions. From the domains of initial conditions which are domains of attraction, probabilities of occurrence of the stable solutions can be calculated. The described dynamical behaviour is studied in the present paper using two examples. To introduce the basic phenomena, the well-known academic Duffing oscillator with harmonic excitation is considered. Domains of attraction are shown for the two stable solutions and the probability of occurrence of the two solutions in the case of equally distributed initial conditions is calculated. The main example to be considered in this paper is the railway wheelset, which is known to show (depending on the nonlinearities in the model) a subcritical Hopf bifurcation depending on the vehicle speed. In that case a range of speeds exists, where a stable trivial solution and a stable limit cycle solution coexist in addition to an unstable limit cycle solution. Again domains of attraction of the two stable solutions are calculated and their dependence on the vehicle speed is shown. The probabilities of occurrence of the two stable solutions are calculated, where special interest is given to the probability of occurrence of the undesired limit cycle solution.