| Summary: | Eigenvalue problems are of immense interest and play a pivotal role not only in many fields of engineering but also in pure and applied mathematics. An in-depth understanding of this class of problems is a pre-requisite for vibration and buckling analyses of structures. Design optimization of structures to prevent failure due to instability (bucking) and vibration introduces the problem of determining optimal physical parameters such that load carrying capacity or the fundamental natural frequency is maximized. A classical example of such problems is the Lagrange problem of determining the shape of the strongest column against buckling. The primary objective of this research is to develop discrete models for column buckling and to solve the problem of finding the strongest column by applying fundamental principles of optimization.
The physical parameters of optimal discrete link-spring models, which maximize the buckling loads, are reconstructed. It is shown that the optimal system can be determined recursively by using a one parameter iterative loop. The mathematical problem of determining parameters of an affine sum such that the system has extremum eigenvalues was derived and solved. Numerical techniques were developed to aid in the optimization process and were utilized to optimize mass-spring systems. The more complicated problem of finding the shape of the strongest column was also defined as an affine sum by applying finite difference schemes to both the second and fourth order governing differential equations. Optimization techniques and numerical methods were developed to arrive at the shape of the strongest clamped-free and pinned-pinned column. Unimodal solutions of the Lagrange problem were also obtained for the special case where minimum area constraints were given. A mathematical model for columns on elastic foundation also was derived and transformed to an affine sum problem. Unimodal solution of shape of the strongest pinned-pinned column on an elastic foundation was obtained. In addition the application in vibration and buckling, it is believed that the optimization principles and numerical methods developed in this research will be applicable in other fields such as optimal control.
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