Stability of a two-mass oscillator moving on a beam supported by a visco-elastic half-space

This paper presents a theoretical study of the stability of a two-mass oscillator that moves along a beam on a visco-elastic half-space. The oscillator and the beam on the half-space are employed to model a bogie of a train and a railway track, respectively. Using Laplace and Fourier integral transforms, expressions for the dynamic stiffness of the beam are derived in the point of contact with the oscillator. It is shown that the imaginary part of this stiffness can be negative thereby corresponding to so-called negative damping. This damping can destabilize the oscillator leading to the exponential growth of the oscillator's displacement. The instability zone corresponding to such behavior is found in the space of the system's parameters with the help of the D-decomposition method. A parametric study of this zone is carried out with the emphasis on the effect of the material damping in the half-space and the viscous damping in the oscillator. It is shown that a proper combination of these damping mechanisms stabilizes the system effectively. An attempt is made to construct a one-dimensional foundation of the beam so that the instability zone predicted by the resulting one-dimensional model would coincide with that obtained from the original three-dimensional model. It is shown that such foundation can be constructed but its parameters are ambiguous and cannot be determined a-priori, without tuning the instability zone. Therefore, it is concluded that one-dimensional models should not be used for the stability analysis of high-speed trains.