Optimum vibrating shapes of beams and circular plates

Optimum vibrating shapes of beams and circular plates, having piecewise linear variation in thickness, are treated herein. The study is concerned with two problems pertaining to (i) the fundamental mode of lateral vibration of beams and (ii) the fundamental mode of axisymmetric vibration of circular plates. These are (a) to find the best shape of the structure which would provide the highest elevation of this fundamental frequency, keeping the volume constant, and (b) to find the minimum volume and shape of the structure for a given minimum allowable fundamental frequency. A numerical procedure incorporating the finite element method and an iterative optimization technique has been used. This has enabled various boundary conditions to be conveniently treated in the analysis. Results indicate that often a very large elevation (>100%%) in the fundamental frequency for volume constraint, and a considerable saving (>50%) in material for frequency constraint, are possible by merely altering the shape. Moreover, the number of slopes and the allowable minimum thickness influence the results.