Natural Frequencies of Longitudinal Oscillations for a Nonuniform Variable Cross-Section Rod / V. I. Gorbachev. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2016. № 1. P. 31-39 [Moscow Univ. Mech. Bulletin. Vol. 72, N 2, 2017. P. 7-15].
The natural frequencies of longitudinal oscillations of a rod such that its Young's modulus, the density, and the cross-sectional area are functions of the longitudinal coordinate are analyzed. For solving the corresponding problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution to the accompanying equation with constant coefficients. Frequency equations are derived in the form of rapidly converging Leibniz series for three types of boundary conditions. For these cases the frequency zeroth-approximation equations are given to quickly find the lowest natural frequencies with an adequate accuracy.
Key words: mechanics of deformable solids, elasticity, dynamic problems, nonuniform rods of variable cross section, averaging methods.