Analytical and experimental studies on out-of-plane dynamic parametric instability of a circular arch under a vertical harmonic base excitation

Abstract

The out-of-plane dynamic stability of an arch under a vertical periodical base excitation is investigated by using both analytical and experimental methods. When the excitation frequency is near twice the out-of-plane natural frequency of the arch, an out-of-plane divergent vibration is induced, which will eventually leads to an out-of-plane dynamic parametric instability of the arch. Firstly, Hamiltonian principle is employed to derive the in-plane dynamic equations, and analytical solutions for the dynamic axial compressive force and bending moment in pre-stability state are obtained and verified by numerical simulations. Then the mode shape functions for the out-of-plane dynamic instability are analytically determined and the out-of-plane dynamic instability regions are obtained by Bolotin's method. Factors affecting the critical excitation frequencies of the out-of-plane dynamic instability regions, such as the arch's included angle, the slenderness ratio, and the logarithmic decrement, are thoroughly analyzed and verified via the method of multiple scales. Lastly, laboratory tests are carried out for the out-of-plane parametric resonance instability of fix-ended circular arches with various included angles and slenderness ratios. The critical excitation frequencies of dynamic instability regions of the tested arches are determined by increasing and decreasing the excitation frequency, and the experimental results agree with the theoretical counterparts very well. The main contributions of this work include: (1) obtaining an analytical solution of out-of-plane dynamic instability regions for the arch under a vertical harmonic base excitation; (2) applying a newly-designed experimental system to conduct the out-of-plane dynamic instability tests; (3) proposing a method of time domain analytical solution to determine the critical excitation frequencies of the out-of-plane dynamic instability region, and (4) discussing the dynamic instability mechanism for base-excited arches.