Efficient Methods for Structural Optimization With Frequency Constraints Using Higher Order Approximations

Presented herein are four different methods for the optimum design of structures subject to multiple natural frequency constraints. During the optimization process the optimum cross-sectional dimensions of elements are determined. These methods are robust and efficient in terms of the number of eigenvalue analyses required, as well as the overall computational time for the optimum design. A new third order approximate function is presented for the structural response quantities, as functions of the cross-sectional properties, and four different methods for the optimum design are defined based on this approximate function. The main features of the proposed function are that only the diagonal terms of higher order derivative matrices are employed, and these derivatives are established by the available first order derivatives. The first order exact derivatives are obtained from a sensitivity analysis at the previous design points. We show that this approximate function creates high quality approximations of the structural responses, such as the frequencies. Examples are presented and the efficiency and quality of the proposed methods are discussed and compared.