中频振动问题有限元与波强度混合分析方法研究

A high-tech equipment such as a launch vehicle, a missile, an aircraft or a high speed train always has a large variety of components with different material properties, topological structures, dimensions, etc., and meanwhile it would be subjected to a very wide band random loading during its whole operation process. This will result in both low frequency long wavelength and high frequency short wave length deformations and be very sensitive to small random imperfections in the system properties. In this project, this mid-frequency vibration problem of complex systems is investigated based on the framework of the hybrid finite element/statistical energy analysis approach. The vibration of the system is described by nodal displacements and average vibrational energy. The technique of dynamic substructure is introduced in the finite element model to reduce its scale significantly. Non-parametric uncertainties in the finite element subsystems are described by random matrix theory. The pseudo excitation method is developed to solve the random responses. The standard assumption that there is equipartition of energy between the modes of a particular statistical subsystem is relaxed and the directional dependency of the wave intensity in each subsystem is represented by a finite Fourier series based on the wave intensity analysis. A hybrid finite element and wave intensity model, which having better numerical stability and wide applicability, is established for mid-frequency vibration analysis in two opposite ways (from low frequency to high frequency and from high frequency to low frequency), and the corresponding efficient numerical methods are also developed. The outcomes of the project will advance the thorough study of theories and methods for related problems.

运载火箭、导弹、飞机、高速列车等高端技术装备的各个组件在材料、结构和尺寸等方面往往存在较大差异，且在运行中受频带非常宽的随机荷载作用，从而产生低频长波和高频短波变形并存的复杂振动特征，并对微小随机缺陷异常敏感。本项目针对复杂结构系统的这类中频振动问题，基于混合有限元与统计能量分析的基本思想，以节点位移和子系统平均振动能量两类变量描述系统的振动。引入动力子结构技术划分有限元子系统以有效降低求解规模，应用随机矩阵理论建立有限元子系统的非参数不确定性模型，发展虚拟激励法实现有限元子系统随机响应的高效计算；基于波强度分析，放松统计性子系统模态能量均分假设，通过傅里叶级数表示波强度的方向依赖性。从低频至高频和从高频至低频两个不同的方向，拓展中频振动问题分析范围，建立具有良好数值稳定性和广泛适用范围的中频振动问题有限元和波强度混合分析模型，发展高效数值方法，为相关问题基本理论与方法的深入研究打下基础。

针对声振系统的中频振动问题提出了一个新的混合分析手段。基于混合FE-SEA方法的框架，采用边界单元法描述声振系统中声腔的确定性振动，通过声腔和结构耦合界面上的位移和速度协调条件，实现了与使用统计能量分析描述的结构之间的耦合，进而建立了声振系统中频振动的混合BE-SEA求解格式，并给出了声腔内部声压响应功率谱的后处理方案。边界元方法的引入不仅可以有效处理外声场问题，而且降低了模型规模和建模难度，进而提高了建模效率。提出一种改进的混合FE-SEA方法。原混合FE-SEA方法在解决组合结构和声振系统的中频振动问题时需要求逆运算得到混合模型的总动柔度矩阵。而混合模型的总动刚度矩阵无法直接使用基于纯有限元模型提出的快速求逆技术进行求逆运算，且混合模型也无法直接利用传统的动凝聚方法进行缩减，所以总动柔度矩阵的求解会产生巨大的计算量。针对这一问题，本项目首先利用动凝聚技术对确定性子系统降阶，从而实现混合模型的整体自由度规模的缩减；然后引入基于纯有限元模型的快速求逆技术求解动凝聚过程中出现的副自由度动柔度矩阵。动凝聚技术显著缩减了系统的总动刚度矩阵的规模；快速求逆技术仅利用矩阵相乘运算得到副自由度动柔度矩阵，以上两步避免了原混合FE-SEA方法中对于大型复数矩阵的直接求逆运算，显著提高了计算效率。研究了声振系统发生中频振动时的吸声材料拓扑优化问题。采用本项目提出的改进混合分析方法作为响应分析方法，基于SIMP方法构造了人工吸声材料模型，以吸声材料的相对密度为设计变量，在给定吸声材料用量条件下，以最小化声腔内指定点的集合平均声压功率谱为目标函数，建立了声振系统中频振动的吸声材料拓扑优化列式。提出的改进混合分析方法不仅能够准确描述声振系统的中频振动行为，而且具有较高的求解效率，对于需要进行多次响应分析的优化问题，显著缩短了系统分析所需的计算时间。同时，利用直接求导法给出了高效的灵敏度分析方案，进一步节省了优化问题求解的计算成本。