Computing critical velocities in waveguides via multiparameter eigenvalue problems

Abstract

In waveguide dynamics and moving-load problems (e.g., high-speed trains), critical velocities indicate the onset of strong vibration amplification. In systems that are invariant in the direction of motion, these velocities can be identified from dispersion relations as points where the phase and group velocities of a propagating mode coincide. Finding such points indirectly by tracing dispersion curves can be cumbersome and potentially unreliable for multimodal systems with complex branch interactions. We present a direct method for computing critical velocities in such scenarios, specifically in the context of semi-analytical methods. Starting from a polynomial parameter-dependent eigenvalue problem for the wavenumber-frequency relation, incorporating the additional condition of equal phase and group velocities yields a singular polynomial multiparameter eigenvalue problem that can be linearized and solved using established algorithms. The proposed approach enables the simultaneous computation of all critical points without requiring the tracing of dispersion curves. Its performance is demonstrated by several benchmark problems, confirming the accurate and robust identification of critical velocities.