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IssuesArchive of Issues2018-1pp.101-110

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L.D. Akulenko, A.A. Gavrikov, and S.V. Nesterov, "Natural Vibrations of a Liquid-Transporting Pipeline on an Elastic Base," Mech. Solids. 53 (1), 101-110 (2018)
Year 2018 Volume 53 Number 1 Pages 101-110
DOI 10.3103/S0025654418010120
Title Natural Vibrations of a Liquid-Transporting Pipeline on an Elastic Base
Author(s) L.D. Akulenko (Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia; Bauman Moscow State Technical University, ul. 2-ya Baumanskaya 5 str. 1, Moscow, 105005 Russia)
A.A. Gavrikov (Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, gavrikov@ipmnet.ru)
S.V. Nesterov (Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia)
Abstract Flexural free vibrations of an ideal-liquid-transporting pipeline on an elastic base are studied. A numerical-analytical method for finding the pipeline natural frequencies and vibration modes is developed, which permits one to determine the natural frequencies and modes for the case in which the tension or compression (the longitudinal force acting along the pipeline axis), the pipe diameter, and hence the velocity of the incompressible fluid being transported are arbitrary functions of the longitudinal coordinate measured along the pipeline axis. The least natural frequencies are calculated for the case in which the variable elasticity of the base is given by some test functions.
Keywords pipeline vibration, natural frequency, Winkler base
References
1.  L. D. Akulenko, M. I. Ivanov, L. I. Korovina, and S. V. Nesterov, "Basic Properties of Natural Vibrations of an Extended Segment of a Pipeline," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela. No. 4, 119-134 (2013) [Mech. Solids (Engl. Transl.) 48 (4), 458-472 (2013)].
2.  G. W. Housner, "Bending Vibrations of a Pipe Line Containing Flowing Fluid," J. Appl. Mech. 19 (2), 205-208 (1952).
3.  C. D. Jr. Mote, "A Study of a Band Saw Vibrations," J. Franklin Inst. 279 (6), 430-444 (1965).
4.  R. Barakat, "Transverse Vibrations of a Moving Thin Rod," J. Acoust. Soc. Am. 43 (3), 533-539 (1968).
5.  M. P. Paidoussis, Fluid-Structure Interactions. Vol. 1: Slender Structures and Axial Flow (Academic Press, Oxford 2014).
6.  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)].
7.  L. D. Akulenko and S. V. Nesterov, "Flexural Vibrations of a Moving Rod," Prikl. Mat. Mekh. 72 (5), 759-774 (2008) [J. Appl. Math. Mech. (Engl. Transl.) 72 (5), 550-560 (2008)].
8.  L. D. Akulenko, A. A. Gavrikov, and S. V. Nesterov, "Natural Oscillations of Multidimensional Systems Nonlinear in the Spectral Parameter," Dokl. Ross. Akad. Nauk 472 (6), 654-658 (2017) [Dokl. Phys. (Engl. Trans.) 62 (2), 90-94 (2017)].
9.  L. D. Akulenko, A. A. Gavrikov, and S. V. Nesterov, "Numerical Solution of Vector Sturm-Liouville Problems with Dirichlet Conditions and Nonlinear Dependence on the Spectral Parameter," Zh. Vych. Mat. Mat. Fiz. 57 (9), 1503-1516 (2017) [Comp. Math. Math. Phys. (Engl. Transl.) 57 (9), 1484-1497 (2017)].
10.  A. A. Gavrikov, "Numerical Solution of Vector Sturm-Liouville Problems with a Nonlinear Dependence on the Spectral Parameter," AIP Conf. Proc. 1863, 560032 (2017).
11.  A. A. Gavrikov, "An Iterative Solution Approach to Eigenvalue Problems for Linear Hamiltonian Systems and its Application to a Hybrid System Control Problem," in 22st Int. Conf. Methods and Models in Automation and Robotics (MMAR). Miedzyzdroje, Poland, 2017 (IEEE, 2017), pp. 588-593.
12.  V. Ph. Zhuravlev and D. M. Klimov, Applied Methods in Theory of Vibrations (Nauka, Moscow, 1988) [in Russian].
13.  F. Riesz and B. Sz.-Nagy, Lectures in Functional Analysis (Budapest, 1953; Mir, Moscow, 1979).
Received 18 September 2017
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