On the conformity of theoretical models of longitudinal rod vibrations with ring defects experimental data

A. L. Popov ; S. A. Sadovskiy

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 1, pp. 97-110.

Résumé

The paper considers a number of theoretical models for describing longitudinal vibrations of a rod. The most simple and common is based on the wave equation. Next comes the model that takes into account the lateral displacement (Rayleigh correction). Bishop’s model is considered to be more perfect, taking into account both transverse displacement and shear deformation. It would seem that the more perfect the theoretical model, the better it should agree with the experimental data. Nevertheless, when compared with the actually determined experimental spectrum of longitudinal vibrations of the rod on a large base of natural frequencies, it turns out that this is not entirely true. Moreover, the most complex Bishop’s model turns out to be a relative loser. The comparisons were made for a bar with small annular grooves that simulate surface defects, which is considered as a stepped bar. The questions of refinement with the help of experimentally found frequencies of the velocity of longitudinal waves and Poisson's ratio of the rod material are also touched upon.

Détail

Keywords: stepped bar, longitudinal vibrations, Rayleigh correction, Bishop's correction, wave equation, experimental data.

Bibliographie

[1] Strutt Lord Rayleigh J. W., The Theory of Sound (in two volumes), v. 1, Dover Publications, New York, 1877

[2] Love A. E. H., A Treatise on the Mathematical Theory of Elasticity, University Press, Cambridge, 1920 | Zbl

[3] Timoshenko S. P., Vibration Problems in Engineering, D. Van Nostrand, New York, 1955

[4] Bishop R. E. D., “Longitudinal waves in beams”, Aeronautical Journal, 3:4 (1952), 280–293 | DOI

[5] Rao S. S., Vibration of Continuous Systems, John Wiley Sons, New York, 2007

[6] Fedotov I. A., Polyanin A. D., Shatalov M. Yu., Tenkam H. M., “Longitudinal vibrations of a Rayleigh–Bishop rod”, Dokl. Phys., 55 (2010), 609–614 | DOI | Zbl

[7] Marais J., Fedotov I., Shatalov M., “Longitudinal vibrations of a cylindrical rod based on the Rayleigh–Bishop theory”, Afrika Matematika, 26:7–8 (2015), 1549–1560 | DOI | Zbl

[8] Shenderov E. L., Volnovye zadachi gidroakustiki [Wave Problems of Hydroacoustic], Sudostroenie, Leningrad, 1972 (In Russian)

[9] Lebedev I. M., Shifrin E. I., “Solution of the inverse spectral problem for a rod weakened by transverse cracks by the Levenberg—Marquardt optimization algorithm”, Mech. Solids, 54:6 (2019), 857–872 | DOI | DOI | Zbl

[10] Kundt A., “Acoustic Experiments”, London Edinburgh Dublin Philos. Mag. J. Sci., 35:4 (1868), 41–48 | DOI

[11] Nerazrushaiushchii kontrol' i diagnostika [Non-destructive Testing and Diagnostics], Handbook, ed. V. V. Klyuev, Mashinostroenie, Moscow, 2003 (In Russian)

[12] Vibratsii v tekhnike [Vibrations in Engineering], Handbook in 6 volumes, v. 1, Kolebaniia lineinykh sistem [Oscillations of Linear Systems], ed. V. N. Chelomei, Mashinostroenie, Moscow, 1978 (In Russian)

[13] Prochnost', ustoichivost', kolebaniia [Strength, Stability, Vibration], Handbook in 3 volumes, v. 3, ed. I. A. Birger, Ya. G. Panovko, Mashinostroenie, Moscow, 1968 (In Russian)

[14] Handbook of Physical Quantities, ed. I. S. Grigoriev, E. Z. Meilikhov, CRC Press, Boca Raton, 1997

[15] Pisarenko G. S., Yakovlev A. P., Matveev V. V., Spravochnik po soprotivleniiu materialov [Handbook on Strength of Materials], ed. G. S. Pisarenko, Nauk. Dumka, Kiev, 1988 (In Russian)

[16] Akulenko L. D., Nesterov S. V., High-Precision Methods in Eigenvalue Problems and Their Applications, CRC Press, New York, 2004