Optimum Design of Vibrating Cantilevers

We determine the optimum tapering of a cantilever carrying an end mass, i.e., the shape which, for a given total mass, yields the highest possible value of the first fundamental frequency of harmonic bending vibrations in the vertical plane. Three different cases are considered. In the first case, all cross sections are assumed to be geometrically similar. In the second case, the cross sections are assumed to be re, angular and of given width. Finally, we consider a rectangular cross section of given height. This third case is shown to be degenerate in the absence of end mass.