Transverse vibrations of double-tapered cantilever beams with end support and with end mass

The free vibrations of a double-tapered cantilever beam with (1) end support and (2) end mass have been investigated using the Bernoulli-Euler equation. The beam was tapered linearly in the horizontal and in the vertical planes simultaneously with the taper ratio in the horizontal plane equal to that in the vertical plane. A table is presented for the first case from which the fundamental frequency, second, third, fourth, and fifth harmonic can easily be obtained for various taper ratios. A chart, plotted from this table, shows the effect of taper ratio on the various harmonics. For the second case, a table and resulting charts show the effect of taper ratio and ratio of end mass to beam mass on the fundamental frequency and higher harmonies. Although previously presented, the case of the beam with free end is also included for purposes of comparison.