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

Shishkina, E.V., Blekhman, I.I., Cartmell, M.P. and Gavrilov, S.N. (2008) Application of the method of direct separation of motions to the parametric stabilisation of an elastic wire. Nonlinear Dynamics, 54(4), pp. 313-331. (doi: 10.1007/s11071-008-9331-9)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Cartmell, Prof Matthew
• Authors: Shishkina, E.V., Blekhman, I.I., Cartmell, M.P., and Gavrilov, S.N.
• College/School: College of Science and Engineering > School of Engineering > Systems Power and Energy
• Journal Name: Nonlinear Dynamics
• Journal Abbr.: Nonlinear Dynam.
• Publisher: Springer
• ISSN: 0924-090X
• ISSN (Online): 1573-269X
• Published Online: 06 February 2008