Asymptotic Formulas for the Eigenvalues of the Timoshenko Beam1

Asymptotic formulas are derived for the eigenvalues of a free-ended Timoshenko beam which has variable mass density and constant beam parameters otherwise. These asymptotic formulas show how the eigenvalues Žand hence how the natural frequencies. of such a beam depend on the material and geometric parameters which appear as coefficients in the Timoshenko differential equations