Transverse Vibrations of Cantilever Bars of Variable Cross Section

Approximate formulas are derived for upper and lower values of the three-lowest natural frequencies for transverse vibrations of cantilever bars of variable cross section. For cones, truncated cones, wedges, and truncated wedges, as well as bars of constant cross section, the effects of rotatory inertia and shear are taken into account also and compared with the results obtained by neglecting these effects according to the elementary Bernoulli-Euler's theory of bending of beams. Numerical results for the first three natural frequencies are presented in tabular form for different inertia characteristics.