基于H_∞理论的结构鲁棒性分析

H_∞-based structural robustness analysis

基于H∞理论,提出一种定量评价结构鲁棒性的新方法。采用状态空间模型描述结构系统,基于H∞最优,采用系统传递函数的H∞范数作为结构鲁棒性的定量评价指标。对线性系统,分离刚度矩阵的不确定性,给出了H∞结构鲁棒性指标的计算方法;对非线性系统,引入L2性能准则表达鲁棒性。通过单自由度体系和桁架结构,明确了鲁棒性指标的物理意义,分析影响结构鲁棒性的因素。结果表明:基于H∞理论的结构鲁棒性指标代表了结构稳态振动反应的最大振幅与干扰幅值之比,可以反映外部干扰和结构内部的不确定性与结构的响应是否成比例;提高结构整体承载力储备、耗能能力以及关键构件的冗余度可以增强结构的鲁棒性;且H∞鲁棒性指标对结构参数变化较为敏感。

Based on the robust H∞ control theory, a framework for the quantitative assessment of structural robustness is proposed. Structures are described by state space model first. Then following the idea of the H∞ optimization, structural robustness is assessed by H∞ norm of the system transfer function. For linear systems, the uncertainty of stiffness matrix is separated, and the calculation method of the H∞ structural robustness index is given. For nonlinear systems, the L2 performance criterion is introduced to express structural robustness. Through analysis of a single degree freedom system and a truss structure, properties of the H∞ structural robustness index are clarified, and factors affecting structural robustness are studied. The results show that, the H∞ structural robustness index is the ratio of the maximum steady-state vibration amplitude and the interference amplitude, which indicates whether the structural response is proportional to the internal and external uncertainty interference. Improvements of both the overall structure performance like load-carrying capacity and energy dissipation capacity reservations, and the local redundancy of key components can enhance structural robustness. Meanwhile the H∞ structural robustness index is sensitive to structural parameters.