为提升几何大变形条件下的结构非线性动力学系统的求解效率，研究指定频域段的动力学行为，以悬壁板为对象，利用频域本征正交分解（Proper Orthogonal Decomposition,POD）方法研究几何非线性结构动力学降阶问题。壁板的几何非线性刚度基于协同转动（Co-rotational，CR）方法求解，利用POD方法在指定频域范围内以计算快照生成基向量，通过Galerkin方法实现动力学系统降阶，非线性刚度以增量的形式加入外力项，系统的非线性行为通过广义外力的形式体现。通过对悬壁板的频域POD降阶分析与对比，结果表明：（1）对线性系统，频域POD降阶分析精度高，误差均在1%以内，求解时间远小于全阶系统，其中，1阶POD的求解时间不到全阶系统的50%；（2）对于非线性系统，在正弦和阶跃载荷作用下，一阶POD降阶分析误差小于1.5%，三阶误差小于0.5%，计算时间均少于全阶分析时间的75%；（3）对于多点随机激励下的几何非线性动力学，通过频域降阶，保留前六阶POD基向量，可保证降阶系统的分析误差在0.5%以内，且计算时间仅为全阶系统的79%。
Abstract
In order to improve solving efficiency of a structural nonlinear dynamic system under large geometric deformation condition and study its dynamic behavior in a specified frequency range, the proper orthogonal decomposition (POD) method in frequency domain was used to study the dynamic order reduction problem of a geometrically nonlinear structure with a cantilevered plate taken as the study object.The geometrically nonlinear stiffness of the plate was solved using the cooperative rotation (CR) method.POD base vectors were generated with snapshots computed in a specified frequency domain, and Galerkin method was used to realize the order reduction of dynamic system.The nonlinear stiffness was added to the external force term in the form of increment, and the nonlinear behavior of the system was reflectedin the form of generalized external force.The POD order reduction analysis in frequency domain and comparison were done for the cantilevered plate.Results showed that (1) for a linear system, the POD order reduction analysis in frequency-domain has high precision, the error is less than 1%, and its solving time is far less than that for the full order system, the solving time for 1 order POD is less than 50% of that for the full order system; (2) for a nonlinear system, the error of 1 order POD analysis is less than 1.5%, and the error of 3 order POD analysis is less than 0.5%, the solving time for the two cases is less than 75% of that for the full order analysis under sine and step loads; (3) for a geometrically nonlinear dynamic system under multi-point random loads, if the first 6 orders POD base vectors are kept after order reduction in frequency domain, the reduced order system’s analysis error is less than 0.5% and its solving time is just 79% of that for the full order system.
关键词
非线性动力学 /
降阶 /
本征正交分解 /
频域 /
协同转动方法 /
几何非线性
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Key words
nonlinear dynamics /
order reduction /
proper orthogonal decomposition(POD) /
frequency domain /
cooperative rotation method /
geometrically nonlinear
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