Sub-harmonic and simultaneous resonance instability of a thin-walled arch under a vertical base excitation at two frequencies

This paper presents the first attempt to investigate the in-plane sub-harmonic and simultaneous resonance instability of a thin-walled arch under a vertical base-excitation at a combination of two frequencies. When the base excitation reaches a critical value, the arch may experience abrupt sub-harmonic and simultaneous resonance instability in an in-plane antisymmetric mode, leading to unpredictable and catastrophic consequences. To understand the sub-harmonic and simultaneous resonance instability behavior of the arch, its in-plane kinematic equation under a vertical base-excitation with two frequencies is derived by employing Hamilton principle and then solved by utilizing Galerkin's method in conjunction with the fourth-order Runge–Kutta method and finite element method to determine the instability regions. The in-plane sub-harmonic and simultaneous resonance instability mechanisms that have not yet been reported in the existing literature are discussed in detail. Some conclusions on the dynamic instability region are come to by comparison between the single- and two-frequency base-excitation. The in-plane sub-harmonic and simultaneous resonance instability analysis can help optimize the design of a thin-walled arch.