Non-stationary longitudinal vibrations of an electromagnetoelastic rod
UDC
593.3DOI:
https://doi.org/10.31429/vestnik-17-2-57-65Abstract
The coupled non-stationary longitudinal vibrations of electromagnetoelastic rods are investigated. It is assumed that the material of the rod is a homogeneous isotropic conductor. The equations used take into account the initial electromagnetic field, the Lorentz force, Maxwell's equations and the generalized Ohm's law. They are obtained as a special case of the corresponding relations for thin electromagnetoelastic shells under the assumption that the desired functions depend only on the longitudinal coordinate.
Two variants of the rods are considered: the infinite rod under the condition that the desired functions are limited and the finite rod with fixed and isolated end cross sections. In the initial electromagnetic field, only the magnetic field strength differs from zero. The solution is presented in integral form with kernels in the form of Green functions. To determine the kernels, the Laplace transform in time and the Fourier transform or trigonometric series in the longitudinal coordinate are used. Explicit forms of solutions are obtained when the transverse compression is ignored and under the conditions of a quasi-static perturbed electromagnetic field. Examples of calculations are provided.
Keywords:
coupled non-stationary electromagnetoelasticity, rod, longitudinal vibrations, Green functions, Laplace transforms, Fourier transforms, trigonometric seriesReferences
- Chirkunov Y.A.Nelinejnye prodolnye kolebaniya vyazkouprugogo sterzhnya v modeli kelvina [Nonlinear longitudinal vibrations of a viscoelastic rod in the kelvin model]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 2015, vol. 79, no. 5, pp. 717–727. (In Russian)
- Zhukov I.A. Prodolnye kolebaniya sterzhnej primenitelno k udarnym sistemam tekhnologicheskogo naznacheniya [Longitudinal vibrations of the rods as applied to shock systems for technological purposes]. Mashinostroenie i inzhenernoe obrazovanie [Mechanical Engineering and Engineering Education], 2016, no. 1, pp. 40–49. (In Russian)
- Zhu X., Li L. Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticityInternational Journal of Mechanical Sciences, 2017, vol. 133, pp. 639–650.
- Mardonov B.M., Rakhmatov R., Rakhmanov A., Tangirov A. Prodolnye kolebaniya fizicheski nelinejnyh sterzhenevyh sistem [Longitudinal vibrations of physically non-linear rod systems]. Znanie [Knowledge], 2016, no. 1-1 (30), pp. 57–60. (In Russian)
- Soleimani Roody B., Fotuhi A.R., Jalili M.M. Nonlinear longitudinal forced vibration of a rod undergoing finite strain. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2018, vol. 232, no. 12, pp. 2229–2243.
- Pozhalostin A.A., Panshina A.V. Prodolnye i krutilnye kolebaniya odnorodnyh sterzhnej i valov [Longitudinal and torsional vibrations of homogeneous rods and shafts]. M.: MSTU named N.E. Bauman, 2016, 36 p. (In Russian)
- Tomilin A.K., Prokopenko E.V. Prodolnye kolebaniya uprugogo elektroprovodnogo sterzhnya v neodnorodnom magnitnom pole [Longitudinal vibrations of an elastic conductive rod in an inhomogeneous magnetic field]. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Bulletin of Tomsk State University. Mathematics and mechanics], 2013, no. 1 (21), pp. 104–111. (In Russian)
- Sdobnikov A.N., Sdobnikov S.A. Prodolnye kolebaniya pezokeramicheskogo sterzhnya pri vozbuzhdenii elektricheskim polem [Longitudinal oscillations of a piezoceramic rod when excited by an electric field]. Inzhenernyj vestnik [Engineering Bulletin], 2014, no. 12. p. 38. (In Russian)
- Chen W.Q., Zhang C.L.. Exact analysis of longitudinal vibration of a nonuniform piezoelectric rod. Second International Conference on Smart Materials and Nanotechnology in Engineering, 2009, pp. 749307–7. DOI: 10.1117/12.840398
- Vestyak V.A., Tarlakovskii D.V. The model of thin electromagnetoelastic shells dynamics. Proceedings of the second international conference on theoretical, Applied and Experimental Mechanics. Structural Integrity. Springer, Nature Switzerland AG, 2019, pp 254–258.
- Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnyh sredah [Waves in continuous media]. M.: Fizmatlit, 2004, 472 p. (In Russian)
- Prudnikov A.P., Brychkov Y.A., Marichev O.I. Integraly i ryady [Integrals and series]. – M.: Nauka, 1981, 800 p. (In Russian)
- Ditkin V.A., Prudnikov A.P. Spravochnik po operacionnomu ischisleniyu. [Reference on operational calculus]. M.: «Vysshaya shkola», 1965, 467 p. (In Russian)
- Abramovits M., Stigan I. Spravochnik po specialnym funkciyam [Handbook of special functions]. M.: Nauka, 1979, 832 p. (In Russian)
- Vestyak V.A., Gachkevich A.R., Musii R.S., Tarlakovskii D.V., Fedotenkov G.V. Dvumernye nestacionarnye volny v elektromanitouprugih telah [Two-dimensional unsteady waves in electromagnetoelastic bodies]. M.: Fizmatlit, 2019, 288 p.(In Russian)
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