PARAMETRIC RESONANCE INSTABILITY IN CIRCULAR ARCH OF FUNCTIONALLY GRADED MATERIALS UNDER VERTICAL UNIFORMLY DISTRIBUTED PERIODIC LOAD
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摘要: 基于拉格朗日原理和哈密顿原理推导了竖向均布周期荷载作用下的功能梯度圆弧拱的动力平衡方程，运用Bolotin方法得到周期为2T的参数共振不稳定域解析解，并与有限元数值模拟结果进行了对比，验证了理论计算结果的正确性。对比分析了均匀孔隙分布(Type1)、线性孔隙分布(Type2)、二次孔隙分布(Type3)、非线性孔隙分布(Type4)对功能梯度圆弧拱面内参数共振不稳定区域的影响，并针对矢跨比与阻尼比对功能梯度圆弧拱参数共振的影响进行了参数分析。结果表明：四种孔隙分布中，二次孔隙分布(Type3)条件下拱的自振频率显著增加，不稳定域更狭窄。同时，参数共振不稳定域受矢跨比与阻尼比的影响较大，矢跨比增大会造成拱自振频率减小，参数共振不稳定域向低频方向移动。Abstract: 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.
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图 4 矢跨比对四种孔隙分布拱不稳定域的影响
Figure 4. Influence of span ratio on four arch instability domains with pore distribution
表 1 一阶振动频率理论值与有限元结果对比
Table 1 Comparison of first-order vibration frequency theoretical values and finite element results
矢跨比 Type1 Type2 理论/Hz FEM/Hz 理论/Hz FEM/Hz 1/4 13.982 13.459 13.387 12.995 1/6 17.113 16.836 16.385 16.260 1/8 18.517 18.372 17.728 17.747 1/10 19.237 19.165 18.418 18.515 矢跨比 Type3 Type4 理论/Hz FEM/Hz 理论/Hz FEM/Hz 1/4 16.543 16.000 13.641 13.193 1/6 20.249 20.015 16.697 16.507 1/8 21.909 21.841 18.066 18.016 1/10 22.762 22.784 18.769 18.796 
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