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

I. S. Nikitin; Никитин И. С.; N. G. Burago; Бураго Н. Г.; A. D. Nikitin; Никитин А. Д.

doi:10.31857/S003282352302011X

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

Authors: Nikitin I.S.¹, Burago N.G.¹^,2, Nikitin A.D.¹
Affiliations:
1. Institute for Computer Aided Design RAS
2. Ishlinsky Institute for Problems in Mechanics RAS
Issue: Vol 87, No 2 (2023)
Pages: 327-336
Section: Articles
URL: https://journals.rcsi.science/0032-8235/article/view/138861
DOI: https://doi.org/10.31857/S003282352302011X
EDN: https://elibrary.ru/UAJFFL
ID: 138861

Cite item

Full Text

Open Access
Restricted Access

Access granted
Restricted Access

Subscription Access

Abstract
About the authors
References
Supplementary files
Statistics

Abstract

The paper is focused on the problem of natural frequencies and modes determination based on perturbation theory for longitudinal and torsional vibrations in bars with variable cross section. The mechanical properties and cross section geometry of the bar are changing small from the average value with regard to longitudinal coordinate. Based on the theory of small perturbations the analytical solution is obtained for natural frequencies and modes of the stationary harmonic vibrations in bars. The efficiency of the proposed method is supported by comparison and good agreement of the obtain results with a sharp solution for a given cross section profiles. It was established that the approximate solution is working good up to the ratio 2.5–3 between maximum and minimum diameter of cross section. The results of numerical simulations are aimed to estimate the geometry and elastic behavior of the metallic specimens for very high cycle fatigue experimental investigation under axial tension-compression and torsion loadings. The piezoelectric fatigue testing system and procedure is based on stationary vibration excitation in the metallic specimen at the first mode natural frequency.

Keywords

very high cycle fatigue, vibrations in bars, variable cross section, tension-compression, perturbation theory, analytical solution, high frequency tests, piezoelectric fatigue testing system

References

Babakov I.M. Oscillation Theory. Moscow: Nauka, 1965. 560 p.
Akulenko L.D., Gavrikov A.A., Nesterov S.V. Identification of defects in the cross section of the rod by natural frequencies and features of the form of longitudinal vibrations // Mech. Solids, 2019, no. 6, pp. 98–107.
Akulenko L.D., Baydulov V.G., Georgievskiy D.V., Nesterov S.V. Evolution of natural frequencies of longitudinal vibrations of a rod with increasing cross-sectional defect // Mech. Solids, 2017, no. 6, pp. 136–144.
Vatulyan A.O., Bocharova O.B. On the reconstruction of the density and Young’s modulus for an inhomogeneous rod // Acoust. Phys., 2009, vol. 55, no. 3, pp. 275–282.
Pavlov V.P., Nusratullina L.R. Torsional vibrations of a rod of non-constant section // Bull. USATU. Mech. Engng.&Engng. Sci., 2022, vol. 26, no. 1 (95), pp. 22–30.
Khakimov A.G. On natural oscillations of a shaft with a model of an artificial defect // Defectoscopy, 2010, no. 6, pp. 93–98.
Gusev B.V., Saurin V.V. On vibrations of inhomogeneous beams // Don Engng. Gazette, 2017, no. 3.
Pavlov V.P., Nusratullina L.R. Exact solutions of the equation describing transverse vibrations of a bar with variable cross section and their application // Bull. USATU. Mech. Engng.&Engng. Sci., 2019, vol. 23, no. 4, pp. 774–779.
Gusev B.V., Saurin V.V. On free bending vibrations of concrete beams of variable cross section // Industrial&Civil Constr., 2019, no. 8, pp. 93–98.
Vatulian A.O., Osipov A.V. On one approach to determining the parameters of a defect in a beam // Defectoscopy, 2014, no. 11, pp. 37–47.
Ilgamov M.A., Khakimov A.G. Diagnosis of damage to a notched cantilever beam // Defectoscopy, 2009, no. 6, pp. 83–89.
Lebedev I.M., Shifrin E.I. Identification of transverse cracks in a bar by natural frequencies of transverse vibrations // Mech. Solids, 2020, no. 4, pp. 50–70.
Naiphe A.Kh. Introduction to Perturbation Methods. Moscow: Mir, 1984. 535 p.
Bathias C., Paris P.C. Gigacycle Fatigue in Mechanical Practice. N.Y.: Marcel Dekker, 2005. 328 p.