In many mechanical and aerodynamic systems structural strength, stability and lightness are of primary importance. The stability of a structure with a specified support and external loading can be characterized by an 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 the system. The problem of designing an optimal structure that can support the largest load is more involved. It requires the determination of structural parameters, e.g., the variable cross-sectional area of the structure, which maximize the least eigenvalue of the system. An important example is due to Lagrange, who attempted to determine the shape of the strongest column with a specific length and mass. This apparently simple problem remained unsolved for more than one hundred and fifty years. Its analytical solution is still surrounded by controversy. The main thrust of this research is that the standard optimization procedure is by no means the best way to address the structural optimization problem involved. Depending on the failure criterion, there is a certain relation between the structural mode of failure and the physical parameters of the optimal structure. In the Lagrange problem the maximum bending stress in the ideal column is constant along its length. This condition implies that the square of the fundamental buckling mode is proportional to the cube of the cross-sectional area for the optimal column. This relation was vital in determining the exact analytical solution to the problem. It also 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 optimal structure. This relation leads to an eigenvalue problem where elements of the stiffness matrix are known functions of an eigenvector of the system, and (b) Solving the inverse eigenvalue problem of determining the unknown parameters in the stiffness matrix subject to the eigenvector constraint. The project has practical engineering applications in new and emerging technologies related to fabrication of microstructures and their integration in new generation of mechanical components. Another important application of the proposed research is in design of safe, reliable, and lightweight aerospace structures The proposed activity will also contribute to the curriculum development of Machine Design Lab and Stress Analysis. The students will take part in a design competition involving optimal design of structures The 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
