PARAMETRIC RESONANCE INSTABILITY IN CIRCULAR ARCH OF FUNCTIONALLY GRADED MATERIALS UNDER VERTICAL UNIFORMLY DISTRIBUTED PERIODIC LOAD

Based on Lagrange principle and Hamiltonian principle, the dynamic equilibrium equation of a functionally gradient arc arch under a vertical uniformly distributed periodic load is derived. Bolotin method is used to obtain the analytical solution of a parametric resonance instability domain with period 2T. The comparison with the finite element numerical simulation results verifies the correctness of the theoretical calculation results. The uniform impact of pore distribution patterns is investigated, including pore distribution (Type1), linear pore distribution (Type2), secondary pore distribution (Type3), and the porosity distribution of nonlinear (Type4) on the instability region of a functionally graded circular arch. Moreover, the influence of the rise-span ratio and of damping ratio on the resonance parameters of the functionally graded circular arch is further analyzed. It is found that: the second natural vibration frequency of the arch increases significantly under the secondary pore distribution (Type3), and the instability domain becomes narrower. It is also found that the parametric resonance instability region is greatly affected by the rise-span ratio and damping ratio. When the span ratio increases, the natural vibration frequency of the arch decreases, and the parametric resonance instability region moves towards the direction of lower frequency.