Geodesic
Torsional vibrations of a rod with an axial geometric inhomogeneity
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 1, pp. 50-58
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The dynamics torsional vibrations of a rod with an axial torsional rigidity inhomogeneity. Rods of different configurations are widely used for simulating stress-strain state in case of static and dynamic load to the objects of mechanical engineering, construction, biomechanics, etc. This work’s objective is creating of a general approach to building mathematical models of torsional vibrations of rods of variable section. As an object we have considered an elastic rod, the torsional rigidity of which changes according to the power law from the longitudinal coordinate. The dynamic process is described by a wave equation, and is solved using the Fourier method. To make the solving of the boundary-value problems more convenient, special functions based on recurrence relations for the Bessel functions are introduced. Taking into account the orthogonal property of weighted eigenfunctions, an expression for the norm square is obtained. As an example a case of vibrations is considered with sudden application of load to one end of the rod, with the other end of the rod being rigidly fixed. The free end is supposed to be under local inertial load. Expressions were obtained for the torsion angles and torques in the cross-sections of the rod. A comparison was performed for the obtained results of calculation in relative values with a simplified single-mass model of the weightless rod.
Mots-clés : torsional vibrations
Keywords: forced vibrations, variable rigidity rod, elastic rod, wave equation, Fourier method, Bessel functions, natural frequencies, stress, strain. 
@article{VYURM_2019_11_1_a6,
     author = {S. N. Tsarenko},
     title = {Torsional vibrations of a rod with an axial geometric inhomogeneity},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {50--58},
     year = {2019},
     volume = {11},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2019_11_1_a6/}
}
TY  - JOUR
AU  - S. N. Tsarenko
TI  - Torsional vibrations of a rod with an axial geometric inhomogeneity
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2019
SP  - 50
EP  - 58
VL  - 11
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURM_2019_11_1_a6/
LA  - ru
ID  - VYURM_2019_11_1_a6
ER  - 
%0 Journal Article
%A S. N. Tsarenko
%T Torsional vibrations of a rod with an axial geometric inhomogeneity
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2019
%P 50-58
%V 11
%N 1
%U http://geodesic.mathdoc.fr/item/VYURM_2019_11_1_a6/
%G ru
%F VYURM_2019_11_1_a6
S. N. Tsarenko. Torsional vibrations of a rod with an axial geometric inhomogeneity. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 1, pp. 50-58. http://geodesic.mathdoc.fr/item/VYURM_2019_11_1_a6/

[1] Sukharev V.A., “Torsional Vibrations of Shaft with Concentrated Masses”, Transactions of Taurida. Agricultural Science, 2015, no. 4 (167), 48–53 (in Russ.)

[2] Donchenko M.A., Zhuravlyov Yu.N., Ivanov A.N., Semyonov S.N., “Parametrically-Excited Torsional Vibrations in One-Module Construction of Rbeehs with Lever-And-Cam Motion Converter”, Scientific and Technical Volga region Bulletin, 2012, no. 3, 114–118 (in Russ.)

[3] Vatul'yan A.O., Denina O.V., “One method of determining the elastic properties of inhomogeneous solids”, Journal of Applied Mechanics and Technical Physics, 53:2 (2012), 266–274 | DOI | MR | Zbl

[4] Khakimov A.G., “On the natural vibrations of a shaft with an artificial defect”, Russian Journal of Nondestructive Testing, 46:6 (2010), 468–473 | DOI

[5] Khakimov A.G., Satyev E.I., “On Drill-String Natural Torsional Vibrations”, Oil and Gas Business, 2014, no. 6, 120–153 | DOI | Zbl

[6] Belyaev A.B., Itbaev V.K., “Torsional Vibrations Calculation of the Shaft Line”, Vestnik transporta Povolzhya, 2009, no. 3, 46–49 (in Russ.)

[7] Shevchenko F.L., Ulitin G.M., Pettik Yu.V., “Torsional impact with free oscillations in drilling rigs”, Progressivnye tekhnologii i sistemy mashinostroeniya, 1999, no. 8, 244–247 (in Russ.)

[8] Shevchenko F.L., Ulitin G.M., “On the types of torsional impacts that occur during the operation of drilling rigs, and how to eliminate them”, Sovershenstvovanie tekhniki i tekhnologii bureniya skvazhin na tverdye poleznye iskopaemye, 24, UGGA, Ekaterinburg, 2001, 132–138 (in Russ.)

[9] Erofeev V.I., Lampsi B.B., “Mathematical model of elastic thin-walled bar committing torsional oscillations in the presence of nonlinearity and warping”, Privolzhsky Scientific Journal, 2014, no. 2, 14–17 (in Russ.)

[10] Kravchuk A.S., Kravchuk A.I., Tarasyuk I.A., “Torsional vibrations of round composite and rheologically active rods”, Proceedings of the IV International Scientific and Practical Conference “Applied Problems of Optics, Informatics, Radiophysics, and Condensed Matter Physics” (Minsk, May 11–12, 2017), Izdatel'stvo Instituta prikladnykh fizicheskikh problem imeni A.N. Sevchenko Publ., Minsk, 2017, 258–260 (in Russ.)

[11] Birger I.A., Panovko Ya.G. (eds.), Strength. Sustainability. Oscillations, Reference in 3 volumes, v. 3, Mashinostroenie Publ., M., 1968, 567 pp. (in Russ.)

[12] Beridze S.P., “Free torsional vibrations of the tapered rod”, Vestnik Orenburgskogo gosudarstvennogo universiteta, 1999, no. 3, 104–107 (in Russ.)

[13] S.K. Biswas, “Note on the torsional vibration of a thin beam of varying cross-section”, Pure and Applied Geophysics, 79:1 (1970), 18–21 | DOI

[14] Ulitin G.M., Tsarenko S.N., “Longitudinal Vibrations of Elastic Rods of Variable Cross-Section”, International Applied Mechanics, 51:1 (2015), 102–107 | DOI | MR

[15] Kamke E., Differentialgleichungen: Lösungsmethoden und Lösungen, Chelsea Pub. Co., New York, 1959 | MR

[16] Watson G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944, 804 pp. | MR | Zbl

[17] Kiselev V.A., Structural mechanics. Special course (dynamics and stability of facilities), Stroyizdat Publ., M., 1964, 332 pp. (in Russ.)

[18] Shevchenko F.L., Dynamics of elastic rod systems, OOO “Lebed'” Publ., Donetsk, 1999, 268 pp. (in Russ.)