In-plane simultaneous resonance instability behaviors of a fixed arch under a two-frequency radial uniformly distributed excitation

This paper delves into a theoretical and numerical exploration of the in-plane simultaneous resonance instability behaviors exhibited by a fixed arch subjected to a two-frequency radial uniformly distributed excitation. An intriguing and previously overlooked phenomenon of simultaneous resonance instability in fixed arches has been identified—specifically, the coexistence of superharmonic resonance and combination subharmonic resonance. The in-plane governing equation of motion was derived using Hamilton's principle and subsequently decoupled through the application of the Galerkin method. Employing the method of multiple scales, boundary excitation frequencies at varying amplitudes were determined, establishing dynamic instability zones and distinguishing between stable and unstable regions of the arch. These findings were further validated using both the finite element method and the fourth-order Runge-Kutta technique. The investigation brought to light a fascinating phenomenon wherein the concurrent manifestation of both 2-order superharmonic resonance and 1/2-order combination subharmonic resonance occurs when the sum of two excitation frequencies approaches twice the natural frequency of the arch, with one of them being approximately half of the natural frequency.