Natural Frequencies and Modes of Longitudinal and Torsional Vibrations in Bars with Variable Cross Section

The problem of determining the frequency and mode of natural longitudinal or torsional vibrations for a bar of variable cross section is considered on the basis of perturbation theory. It is assumed that the elastic properties and cross-sectional area of a straight bar change quite slowly and slightly deviate from some average values along the longitudinal coordinate. Analytical formulas for corrections to the natural frequencies and modes of stationary harmonic oscillations of the rod are obtained using the method of asymptotic expansions over a small parameter. The performance of the formulas is verified by comparison with exact solutions for some dependences of the cross-sectional area on the longitudinal coordinate. It is shown that the approximate formulas work well even for bars with the ratio of the maximum and minimum cross-section radius reaching 2.5-3. Numerical calculations are focused on evaluating the geometric and elastic properties of samples for experimental studies of the fatigue strength of metal alloys under high-frequency cyclic loading in tension-compression and torsion. Piezoelectric testing machines for carrying out such high-frequency tests are based on the general principle of resonant loading of corset samples with a frequency of about 20 kHz.