Instability of a moving mass suspended electromagnetically from a periodically supported beam at high speed

Abstract

This study addresses the dynamic stability of a moving mass suspended electromagnetically from a flexible beam that is supported periodically by discrete elastic springs. The stability is generally determined by the interaction of the wave-induced and electromagnetic instability mechanisms. Both are related to a potentially destabilizing force: the controlled electromagnetic force and the reaction force of the guideway (beam-foundation system). The former is destabilizing if the control is inappropriate, and the latter when sufficiently energetic anomalous Doppler waves are excited in the guideway that feedback energy into the vehicle vibration. Using a generalization of Hill’s method, the stability boundary is determined in the plane of electromagnetic-control parameters. The obtained boundary is roughly triangular, like for the equivalent non-periodic system. The left, straight boundary marks the emergence of a divergence instability. The right boundary generally marks the emergence of an oscillatory (flutter-type) instability, but specific, elliptical indentations are related to parametric resonances. The divergence instability is always electromagnetics induced, but the oscillatory instability and parametric resonances can be either wave or electromagnetics induced, although the latter are often electromagnetics induced. Wave-induced instability takes place mostly for large speeds and only for small values of the control parameters. The stability boundary locally bends back there, reducing the size of the stable zone considerably. Next to the T and 2T parametric-resonance indentations, the right boundary has a significant amorphous indentation compared to that of the non-periodic system. Furthermore, the 2T parametric resonance ellipse is very significant in size when the inhomogeneity of the periodic guideway is relatively strong. Interestingly, the amorphous indentation is related to the occurrence of an evanescent wave in the periodic guideway, but parametric resonance appears to be not uniquely related to a single wave type. Although the current study is fundamental in nature, the findings do pave the way towards the design of safe and cost-effective Maglev and Hyperloop infrastructure as well as of electromagnetic-suspension controllers. We emphasize that the wave-induced instability mechanism, and more generally speaking the influence of the periodic guideway, is also relevant in the context of other (than the simple PD) control strategies as well as for different Maglev and Hyperloop suspension/levitation systems such as the electrodynamic, the hybrid and the superconducting magnet suspensions.