Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
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IssuesArchive of Issues2015-2pp.227-231

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L.D. Akulenko, D.V. Georgievskii, and S.V. Nesterov, "Transverse Vibration Spectrum of a Part of a Moving Rod under a Longitudinal Load," Mech. Solids. 50 (2), 227-231 (2015)
Year 2015 Volume 50 Number 2 Pages 227-231
DOI 10.3103/S0025654415020120
Title Transverse Vibration Spectrum of a Part of a Moving Rod under a Longitudinal Load
Author(s) L.D. Akulenko (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia,
D.V. Georgievskii (Lomonosov Moscow State University, Moscow, 119992 Russia,
S.V. Nesterov (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia)
Abstract We study the natural transverse vibrations of a constant-length part of a rectilinear thin rod moving along a neutral line in unstrained state. The displacement occurs between two fixed coaxial rod guides (clamps) and the distance between them is equal to the length of the vibration part of the rod. Moreover, it is assumed that a constant longitudinal force acts along the neutral line, and two significantly different cases are distinguished: the force extends the rod, and the force compresses the rod.

The vibration process is described by a nonself-adjoint boundary value problem. For arbitrary values of the rod displacement velocity and longitudinal forces, the natural frequencies of possible vibration modes are determined and analyzed by a numerical-analytical procedure with a prescribed accuracy. The global properties of the spectrum depending on the velocity, the longitudinal force, and mode number are determined. For higher-order modes, the domains are discovered where the frequencies nonuniquely depend on the rod motion velocity and the value and direction of the longitudinal force and where the lower vibration frequencies can be absent as the rod motion velocity increases. It is shown that the picture of partial vibrations from the standpoint of an immovable observer cardinally differs from the commonly known picture for the immovable rod.

The obtained results are interesting for studying vibrations of various elements of moving elastic media, including the case of systems with moving boundaries. These results can also have technical applications in problems of dynamics and strength of devices, machines, and mechanisms in textile industry in the filament and rope manufacturing and in metallurgy, in particular, in rolling of metallic rods and strips, in wire drawing, and in production of plastic articles and paper rolls. The developed technique for calculating the natural frequencies and shape modes can be used to analyze the transverse vibrations of parts of transport pipelines with rapidly flowing liquids.
Keywords rod motion, longitudinal force, flexural vibration, nonself-adjoint problem, Ferrari formulas, secular equation, natural frequencies
1.  V. A. Svetlitskii, Mechanics of Rods, Vol. 2 (Vysshaya Shkola, Moscow, 1987) [in Russian].
2.  C. D. Mote, "A Study of Band Saw Vibrations," J. Franklin Inst. 279 (6), 430-444 (1965).
3.  R. Barakart, "Transverse Vibrations of a Moving Thin Rod," J. Acoust. Soc. Amer. 43 (3), 533-539 (1968).
4.  S. V. Nesterov and L. D. Akulenko, "Spectrum of Transverse Vibrations of a Moving Rod," Dokl. Ross. Akad. Nauk 420 (1), 50-54 (2008) [Dokl. Phys. (Engl. Trans.) 53 (5), 265-269 (2010)].
5.  L. Kong and R. G. Parker, "Approximate Eigensolutions of Axially Moving Beams with Small Flexural Stiffness," J. Sound Vibr. 276, 459-469 (2004).
6.  L. D. Akulenko and S. V. Nesterov, "Flexural Vibrations of a Moving Rod," Prikl. Mat. Mekh. 72 (5), 766-781 (2008) [J. Appl. Math. Mech. (Engl. Transl.) 72 (5), 550-560 (2008)].
Received 07 April 2014
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