Optimal sizing of vibrating Bernoulli columns accounting for axial compression and self‐weight

We investigate the effect of axial compression on the optimal design of columns for the maximization of the fundamental frequency of vibration. The compression may be caused by a force applied at the columns' tip or by a load distribλuted along its axis, which may act either independently or simultaneously. We discuss the influence of these contributions on the optimality conditions, and show how the optimal beam design, and the corresponding frequency gain, change with the level of compression. We also discuss the indirect effect of frequency optimization on the critical load factors associated with the tip (λ_p) and distributed (λ_Q) loads. Finally, we provide quantitative results for the optimal design problem parameterized by the triple (λ_p, λ_Q, Ω², ) of buckling and dynamic eigenvalues.