Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

E.V. Shishkina, I.I. Blekhman, M.P. Cartmell, S.N. Gavrilov

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.
Original languageEnglish
Pages (from-to)313-331
Number of pages19
JournalNonlinear Dynamics
Volume54
Issue number4
DOIs
Publication statusPublished - 31 Dec 2008

Fingerprint

Stabilization
Vertical
Wire
Support Point
Motion
Equations of motion
Mechanics
Distributed Systems
Explicit Formula
Equations of Motion
Stiffness
Vibration
Unstable
Acoustic waves
Boundary conditions
Methodology
Sound

Keywords

  • inverted pendulum
  • parametric excitation
  • stability of column
  • vibrational stabilization

Cite this

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Application of the method of direct separation of motions to the parametric stabilization of an elastic wire. / Shishkina, E.V.; Blekhman, I.I.; Cartmell, M.P.; Gavrilov, S.N.

In: Nonlinear Dynamics, Vol. 54, No. 4, 31.12.2008, p. 313-331.

Research output: Contribution to journalArticle

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