Optimal sizing of 1D vibrating 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 vibration frequency. The compression may be due to a force at the columns' tip or to a load distributed 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 drastically change with the level of compression. We also discuss the indirect effect of frequency optimization on the critical load factors for the tip ($\lambda_{P}$) and distributed ($\lambda_{Q}$) loads. Finally, we provide some quantitative results for the optimal design problem parametrized by the triple ($\lambda_{P}$, $\lambda_{Q}$, $\Omega^{2}$) of buckling and dynamic eigenvalues.