Frequencies of Longitudinal Vibration for a Slender Rod of Variable Section

The natural frequencies of a slender, homogeneous, fixed-free rod of variable section are studied in a one-dimensional theory. The rod is uniform at the fixed end for a certain distance L₁, and then tapers for a distance L₂ so that the cross-sectional area varies linearly. The frequency equation is derived and its first five zeros are tabulated for several ratios of L₁ to L₂, and various tapers. The first few terms of asymptotic expansions for the roots of the frequency equation are presented and discussed.