Structural Optimization and Stability

在许多机械和空气动力系统中，结构强度、稳定性和轻量化是首要的。结构在特定支承和外载作用下的稳定性可以用特征值问题来刻画，其解决定了系统的非平凡平衡构型，而施加在结构上的最大外荷载则由系统的最小特征值决定。如何设计出能够承受最大荷载的最优结构是一个较为复杂的问题。它要求确定结构参数，例如结构的可变截面积，使系统的最小特征值最大化。一个重要的例子是拉格朗日，他试图确定具有特定长度和质量的最坚固的柱子的形状。这个看似简单的问题150多年来一直没有得到解决。它的分析解决方案仍然存在争议。本研究的主旨在于，标准的优化程序绝不是解决所涉及的结构优化问题的最佳方法。根据破坏准则，结构的破坏模式与最优结构的物理参数之间存在一定的关系。在拉格朗日问题中，理想柱的最大弯曲应力沿其长度方向是恒定的。这一条件意味着基本屈曲振型的平方与最佳柱的截面积的立方成正比。这一关系对于确定问题的确切解析解至关重要。它在本文提出的新方法中也起着重要的作用，该方法包括两个阶段：(A)确定破坏模式-形状与优化结构的物理参数之间的关系。这种关系导致了一个特征值问题，其中刚度矩阵的元素是系统特征向量的已知函数，以及(B)求解在特征向量约束下确定刚度矩阵中未知参数的逆特征值问题。该项目在与微结构制造及其在新一代机械部件中的集成相关的新技术中具有实际工程应用。建议的研究的另一个重要应用是安全、可靠和轻量化的航空结构设计。建议的活动也将有助于机械设计实验室和应力分析课程的发展。学生将参加一项涉及结构优化设计的设计竞赛该项目将通过以下方式支持弱势少数族裔学生：(A)直接让他们参与研究；(B)激发他们继续深造的兴趣

AbstractIn many mechanical and aerodynamic systems structural strength, stability and lightness are of primaryimportance. The stability of a structure with a specified support and external loading can be characterized byan eigenvalue problem whose solution determines the non-trivial equilibrium configurations of the system.The maximum external load that can be imposed on the structure is determined by the least eigenvalue of thesystem. The problem of designing an optimal structure that can support the largest load is more involved. It requiresthe determination of structural parameters, e.g., the variable cross-sectional area of the structure, whichmaximize the least eigenvalue of the system. An important example is due to Lagrange, who attempted todetermine the shape of the strongest column with a specific length and mass. This apparently simple problemremained unsolved for more than one hundred and fifty years. Its analytical solution is still surrounded bycontroversy. The main thrust of this research is that the standard optimization procedure is by no means the best way toaddress the structural optimization problem involved. Depending on the failure criterion, there is a certainrelation between the structural mode of failure and the physical parameters of the optimal structure. In theLagrange problem the maximum bending stress in the ideal column is constant along its length. This conditionimplies that the square of the fundamental buckling mode is proportional to the cube of the cross-sectional areafor the optimal column. This relation was vital in determining the exact analytical solution to the problem. Italso plays a major role in the new methodology proposed here, which consists of two stages:(a) Determining the relation between the mode-shape of failure and the physical parameters of the optimalstructure. This relation leads to an eigenvalue problem where elements of the stiffness matrix are knownfunctions of an eigenvector of the system, and(b) Solving the inverse eigenvalue problem of determining the unknown parameters in the stiffness matrixsubject to the eigenvector constraint.The project has practical engineering applications in new and emerging technologies related to fabricationof microstructures and their integration in new generation of mechanical components. Another importantapplication of the proposed research is in design of safe, reliable, and lightweight aerospace structuresThe proposed activity will also contribute to the curriculum development of Machine Design Lab andStress Analysis. The students will take part in a design competition involving optimal design of structuresThe project will support disadvantaged minority students by (a) directly involving them in the research,and (b) stimulating their interest in advancing themselves to pursue graduate study