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.

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