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IssuesArchive of Issues2006-2pp.81-86

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S. A. Agafonov and G. A. Shcheglov, "On the instability of a free elastic beam with non-linear internal viscosity under a follower force," Mech. Solids. 41 (2), 81-86 (2006)
Year 2006 Volume 41 Number 2 Pages 81-86
Title On the instability of a free elastic beam with non-linear internal viscosity under a follower force
Author(s) S. A. Agafonov
G. A. Shcheglov
Abstract Stability of elastic bodies and structural elements under non-conservative loads is one of the subject areas in the mechanics of deformable solids and stability theory which has been of permanent interest through the entire past century [1-3].

In the literature on dynamic stability, a phenomenon called "Ziegler's paradox" is well known. An infinitesimal viscosity in linear non-conservative systems causes a decrease in the stability boundary by a finite value. Similar phenomena are discussed in detail, e.g., in the review [4]. In [5], the dynamic stability of a beam with non-linear internal viscosity is studied. One end of the beam is clamped, while the other end is subjected to a constant (in magnitude) follower force. It is shown that the effect of decrease in the stability boundary also takes place in the presence of non-linear viscosity.

In the present paper, a free straight beam having non-linear internal viscosity and loaded by a follower force at one end is considered. It is shown that in this case the decrease in the stability boundary takes place as well.
References
1.  V. V. Bolotin, Non-conservative Problems of the Theory of Elastic Stability [in Russian], Fizmatgiz, Moscow, 1961.
2.  A. P. Filin, Applied Mechanics of Solids. Volume 3 [in Russian], Nauka, Moscow, 1981.
3.  N. Kh. Arutyunyan, A. D. Drozdov, and V. B. Kolmanovskii, "Stability of viscoelastic bodies and structural members," in Achievements in Science and Technology. Solid Mechanics [in Russian], Vol. 19, pp. 3-77, VINITI, Moscow, 1987.
4.  A. P. Seyranyan, "Destabilization paradox in the stability problems for non-conservative systems," Uspekhi Mekhaniki [Advances in Mechanics], Vol. 13, No. 2, pp. 89-124, 1990.
5.  S. A. Agafonov and D. V. Georgievskii, "Dynamic stability of a beam with non-linear internal viscosity under a follower force," Doklady RAN, Vol. 396, No. 3, pp. 339-342, 2004.
6.  V. I. Feodos'ev, "On one stability problem," PMM [Applied Mathematics and Mechanics], Vol. 29, No. 2, pp. 391-392, 1965.
7.  L. G. Khazin and E. E. Shnol', Stability of Critical Equilibrium States [in Russian], Center for Biological Studies of the USSR Academy of Sciences, Pushchino, 1985.
Received 25 February 2005
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