Abstract
In this work, we study the nonlinear oscillations of mechanical systems resting on a (unilateral) elastic substrate reacting in compression only. We consider both semi-infinite cables and semi-infinite beams, subject to a constant distributed load and to a harmonic displacement applied to the finite boundary. Due to the nonlinearity of the substrate, the problem falls in the realm of free-boundary problems, because the position of the points where the system detaches from the substrate, called Touch Down Points (TDP), is not known in advance. By an appropriate change of variables, the problem is transformed into a fixed-boundary problem, which is successively approached by a perturbative expansion method. In order to detect the main mechanical phenomenon, terms up to the second order have to be considered. Two different regimes have been identified in the behaviour of the system, one below (called subcritical) and one above (called supercritical) a certain critical excitation frequency. In the latter, energy is lost by radiation at infinity, while in the former this phenomenon does not occur and various resonances are observed instead; their number depends on the statical configuration around which the system performs nonlinear oscillations.
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References
Awrejcewicz, J., Krysko, V.A.: Introduction to Asymptotic Methods. Chapman and Hall Boca Raton, FL (2006)
Callegari, M., Carini, C.B., Lenci, S., Torselletti, E., Vitali, L.: Dynamic models of marine pipelines for installation in deep and ultra-deep waters: Analytical and numerical approaches. In: Proceedings of the AIMETA03, Ferrara, September 9–12, 2003 (CD-ROM)
Celep, Z., Malaika, A., Abu-Hussein, M.: Forced vibrations of a beam on a tensionless foundation. J. Sound Vib. 128(2), 235–246 (1989)
Chin, C.M., Nayfeh, A.: Three-to-one internal resonances in parametrically excited hinged-clamped beams. Nonlinear Dyn. 20, 131–158 (1999)
Couliard, P.Y., Langley, R.S.: Nonlinear dynamics of deep-water moorings. In: Proceedings of the OMAE’01. Rio de Janeiro, Brasil (2001)
Crank, J.: Free and Moving Boundary Problems. Oxford University Press, UK (1984)
Del Piero, G., Maceri, F. (eds.): Unilateral problem in structural analysis. CISM Courses and Lectures, 288 (1985)
Doyle, J.F.: Wave Propagation in Structures. Springer-Verlag (1989)
Kolsky, H.: Stress Waves in Solids. Dover Publications, New York (1963)
Lenci, S., Callegari, M.: Simple analytical models for the J-lay problem. Acta Mechanica 178(1–2), 23–39 (2005)
Lenci, S., Lancioni, G.: An analytical and numerical study of the nonlinear dynamics of a semi-infinite beam on unilateral Winkler soil. In: Proceedings of the World Congress on Computational Mechanics (WCCM7). Las Vegas, NV, July 16–22, 2006 (CD-ROM)
Nayfeh, A.: Perturbation Methods. Wiley-Interscience, New York (1973)
Nayfeh, A., Balachandran, B.: Applied Nonlinear Dynamics. Wiley-Interscience, New York (1995)
Toscano, R.: Un problema dinamico per la piastra su suolo elastico unilaterale. In: Del Piero, G., Maceri, F. (eds.), Unilateral Problems in Structural Analysis. CISM Courses and Lectures, vol. 288, pp. 375–387 (2005) (in Italian)
Weitsman, Y.: On foundations that react in compression only. ASME J. Appl. Mech. 37(4), 1019–1030 (1970)
Wolf, J.P.: Soil-Structure-Interaction Analysis in Time Domain. Prentice Hall, Englewood Cliffs, New Jersey (1988)
Yilmaz, E.: One dimensionale Schrodinger equation with two moving boundaries. ArXiv:math-ph/030206 v2 5 (February 2003)
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Demeio, L., Lenci, S. Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate. Nonlinear Dyn 49, 203–215 (2007). https://doi.org/10.1007/s11071-006-9122-0
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DOI: https://doi.org/10.1007/s11071-006-9122-0