Instability of a train of oscillators moving along a beam on a viscoelastic foundation

Abstract

The paper focuses on the theory of the vibration instability in the train of oscillators uniformly moving along an Euler–Bernoulli beam on a viscoelastic foundation, caused by the anomalous Doppler waves excited in the beam. This theory may be applied to the high-speed trains when the speed exceeds the minimum phase velocity of the elastic waves that they induce in the track. Each oscillator has two masses connected to a system Kelvin–Voigt representing a single wheel of the train and the corresponding suspended mass. The Euler–Bernoulli beam on viscoelastic foundation models the track, including the rail bending stiffness, the inertia of the track and the subgrade viscoelasticity. The wheel/rail Hertzian contact and the possibility of the contact loss are accounted for. The analysis of the dynamic behaviour of the train of oscillators has two distinct sections. For the former one, the linear critical velocity is calculated starting from the roots of the characteristic equation by applying either the D-decomposition method or an iterative one. For the latter one, the nonlinear stability based on the bifurcation theory is analysed. To this end, the nonlinear equations of motion are solved via a new form of the Green's functions method. The above theory has been applied to the particular case of two oscillators moving along a beam on viscoelastic foundation, to point out the instability behaviour of a bogie. The result of the bifurcation analysis is that the oscillators/beam system exhibits a sub-critical bifurcation. The critical velocity of the system is given by the value of the nonlinear critical velocity. As long as the velocity is within the range between the nonlinear and linear critical velocities, the system motion can have two stable behaviours depending on the initial perturbation: the equilibrium points when the perturbation is sufficiently low or, alternatively, an asymmetric limit cycle as periodic attractor. When the velocity of the two oscillators is higher than the value of the linear critical velocity, the motion is a stable limit cycle which can be a periodic or chaotic attractor. The stable limit cycle is described by successive shocks between the low mass of the oscillators and the beam accompanied by the repetitive contact loss at both oscillators. The contact force peaks are very high, mainly at the rear oscillator. In fact, the motion amplitude is higher for the rear oscillator, due to the shock wave effect.