带频率禁区的结构单元尺寸二阶灵敏度动力优化设计

结构在服役期间经常受到风、地震或车载等激振作用,优化调整结构设计参数来设计结构动力特性以使其避开外部激励对应的卓越频率或高能频带,是提升结构安全性的有效手段。本文针对尺寸约束、频率禁区约束和结构重量最小化的结构动力学优化设计问题,基于Kuhn-Tucker条件和频率对变量的灵敏度,采用泰勒二级展开式推导并建立了结构频率对变量的二阶灵敏度单因子迭代计算方法,利用MATLAB平台编制了计算程序。算例表明比一阶灵敏度算法计算效率高、收敛稳定性强,且修正因子在优化全过程中无需调整,操作更为便利,并给出了单因子的合理取值区间。发现并初步论证了在设计变量未全部到达约束上下界时,“重量不再降低”只能作为最优的必要条件,“高阶频率收敛于频率禁区上限” 则应是充要条件,更适合作为收敛判据,其可有力甄别“伪最优”状况。基于数据表征,初步揭示了优化变量的修改主要受频率梯度主导,并发现频率梯度值大小与变量修正幅度大体成反比的变化规律。本研究对结构的抗风、抗震和在役结构的动力加固改造设计均具有重要的理论指导价值和现实意义。

Second-order Sensitivity Dynamic Optimization Design of Structural Element Size with Frequency- prohibited Band

Adjusting structural dynamic characteristics by searching reasonable structural design parameters to avoid the predominant frequencies or high energy bands corresponding to the external excitation for structures subjected to wind, earthquake or vehicle-borne excitation, which can inevitably improve the safety of structures during service. In this paper, an optimal method of structural dynamic design with dimension constraints, single frequency-prohibited band constraint and structural weight minimization is studied. A single factor iteration algorithm of second-order sensitivity of structural frequency to variables based on the Kuhn-Tucker condition and the sensitivity of frequency to variables is deduced and established by Taylor's second-order expansion formulation. The first-order and second-order sensitivity computing programs in terms of the same example are developed based on the platform of MATLAB. The example shows that the optimization results of both algorithms are identical, but the second-order sensitivity algorithm is more efficient and convergent than the first-order algorithm, and the correction factor doesn’t need to be adjusted in the whole process of optimization, which is more convenient for operators. The reasonable range of the single factor is given. It is found and preliminarily demonstrated that" no weight reduction", when all design variables do not reach the upper and lower bounds of the constraints, can only be regarded as the necessary condition for optimum, while "higher-order frequency converges to the upper limit of the frequency-prohibited band " should be a sufficient and necessary condition, which is more suitable as a convergence criterion. The conclusion can effectively discriminate the "pseudo-optimal" situation. Based on data representation in the example, it is preliminarily revealed that the modification of optimization variables is mainly dominated by frequency gradient, and the magnitude of frequency gradient is inversely proportional to the magnitude of variable modification. The work done here has important theoretical guiding value and practical significance for the design of wind resistance, earthquake resistance, dynamic reinforcement or reconstruction of structures in service.