Keywords: suspended cable, moving oscillators, dynamic interaction, Galerkin method, mode-acceleration method, quasi-static solution, slope discontinuity.

In many engineering applications, such as aerial cableways, ski lift or tramways. cables are used as a means of transportation. For this reason, the analysis of cable vibrations under moving sub-systems may be of great interest for design purposes. Indeed, the dynamics of stretched strings subject to moving loads has attracted significant attention in the literature [1,2] while, to the authors' knowledge, comparatively fewer studies have focused on the response analysis of sagging cables to moving loads [3-5].

This chapter deals with the dynamic analysis of in-plane vibrations of flat-sag suspended cables carrying a stream of oscillators moving with an arbitrary time-law. An efficient procedure recently proposed by the authors [5] is revisited. The suspended cable is modelled as a mono-dimensional homogeneous elastic continuum fully accounting for geometrical nonlinearities. The equations of motion of the coupled cable-moving oscillators system are derived. In particular, ignoring the longitudinal inertia forces and applying a standard condensation procedure, the motion equations of the cable is reduced to a unique nonlinear integro-differential equation in the vertical cable vibrations. The ordinary differential equations ruling the response of the moving oscillators are expressed in terms of absolute displacements. In a first stage, an approximate solution is pursued by means of the Galerkin method approximating the cable displacement as a series expansion in terms of appropriate basis functions. The discretization yields a set of coupled nonlinear ordinary differential equations in the generalized coordinates of the cable and absolute displacements of the oscillators which is characterized by a constant diagonal mass matrix and sparse time-dependent stiffness and damping matrices, the latter matrix also being symmetric. Then, in order to overcome the inability of the traditional Galerkin method accurately to reproduce the kinks and abrupt changes of cable configuration at the contact points with the moving oscillators, following the well-known "mode-acceleration" method [6], a correction term representing the "quasi-static" contribution of the truncated high frequency modes is added to the conventional series expansion [5].

Numerical results concerning a cable carrying a pair of oscillators demonstrate that, despite continuous basis functions being employed, the introduction of the quasi-static correction enables the kinks and abrupt changes in cable configuration at the interface with the moving sub-systems to be captured with very few terms.

The versatility and computational efficiency make the method presented a potentially useful tool for design purposes.

[1]

H.S. Tzou, L.A. Bergman, "Dynamics and control of distributed systems", Cambridge University Press, New York, 1998.

[2]

L. Frýba, "Vibration of solids and structures under moving loads", Thomas Telford, London, 1999.

[3]

Y.M. Wang, "The transient dynamics of a cable-mass system due to the motion of an attached accelerating mass", International Journal of Solids and Structures", 37, 1361-1383, 2000. doi:10.1016/S0020-7683(98)00293-5

[4]

M. Al-Qassab, S. Nair, J. O'Leary, "Dynamics of an elastic cable carrying a moving mass particle", Nonlinear Dynamics, 33, 11–32, 2003. doi:10.1023/A:1025558825934

[5]

A. Sofi, G. Muscolino, "Dynamic analysis of suspended cables carrying moving oscillators", International Journal of Solids and Structures, 44(21), 2007, 6725-6743. doi:10.1016/j.ijsolstr.2007.03.004

[6]

N.R. Maddox, "On the number of modes necessary for accurate response and resulting forces in dynamic analysis", Journal of Applied Mechanics (ASME), 42, 516-517, 1975. doi:10.1115/1.3423622

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