Dynamic response and wave motion of a periodically supported beam under an ultra-high-speed load: Wave dispersion and critical velocities

Abstract

Vacuum tube transportation system is an attractive transportation mode due to the ultra-high-speed (over 1000 km/h) it can reach. Most vacuum tube transportation systems, e.g., the Hyperloop operate in elevated tube structure on periodical supports. When the speed of the vehicle reaches certain critical values, the periodical supporting structure experiences resonance-like response whose vibrational magnitude is significantly amplified. The amplification of the response has an adverse impact on the safety of the vehicle operation and severability of the supporting structures. This work aims to boost the fundamental understanding of the underlying physical mechanism of wave motions and critical velocities in the “moving load on periodical structure” problem utilizing an ultra-high-speed Hyperloop-type transportation system as an example. In this work, an ultra-high-speed moving load on a periodically supported beam is used to model the dynamic response of the supporting structure under an ultra-high-speed travelling vehicle. The steady-state response of the periodically supported beam is solved analytically and illustrated for sub- and trans-critical speeds. The dispersion relation is obtained while deriving the steady-state response. The relation between wave propagation in a periodical structure and critical speeds of the moving load on it is established. It is shown that the critical speeds could be within the operational speed range of ultra-high-speed transportation system. The number and magnitude of critical speeds are closely related to wave motions in the periodically supported structure. It is found that the primary critical speeds associated with largest beam responses may not exist, depending on the relative flexibility of the supports. The optimal structural parameters are associated with wave motion transition (between Local Resonance and Bragg Scattering) in the supporting structure. With this simplified model, wave motions in a periodical structure and critical speeds of the moving load on it is explained in a general physical sense. This work provides theoretical basis for the structural design and operation of the ultra-high-speed transportation system, as well as other engineering applications in which ultra-high-speed moving load is involved.