Keywords: suspended cable, moving masses, convective acceleration, series expansion, mode-acceleration method, quasi-static solution, slope discontinuity.

Cables are used in many engineering applications, such as electrical and optical signal transmission lines, tethers and umbilicals, aerial cableways, cable supported bridges etc. It is well-known, however, that due to their inherent flexibility, light weight and low structural damping, cables are prone to large amplitude oscillations, which may impair their performances. For this reason since the 18th century many investigations have focused on the dynamics of suspended cables [1].

The addition of attached masses, simulating sensors, buoyancy elements, instruments (e.g. hydrophones or cameras), greatly complicates the description of cable motion [2] which cannot be accomplished by directly applying the procedures developed for bare cables, i.e. those without masses. An attached mass, in fact, produces a singularity which makes a cable deformed profile which can no longer be described by a smooth continuous curve. In many engineering applications, such as aerial cableways or tramways, the analysis of cable vibrations is even more difficult. In fact, the dynamics of a nonlinear distributed system (the cable) coupled to moving masses or oscillators (cabins or payloads) has to be investigated. The dynamics of stretched strings has attracted significant attention in the past [3], while, to the authors' knowledge, comparatively fewer studies have focused on the response of sagged cables to moving loads [4,5,6].

This paper deals with nonlinear in-plane vibrations of suspended cables with small sag-to-span ratios carrying an arbitrary number of moving masses. The cable is modelled as a mono-dimensional elastic continuum, fully accounting for geometrical nonlinearities. The interaction between the moving masses, regarded as rigid bodies, and the supporting structure is appropriately described by retaining the convective acceleration terms. The equations governing nonlinear in-plane vibrations of the cable-mass system are derived by applying the extended Hamilton's principle. Neglecting the longitudinal inertia forces of both the cable and the masses, and applying a standard condensation procedure, the equations of motion are reduced to a unique integro-differential equation in the vertical displacement component. The spatial dependence is eliminated by the Galerkin method representing the solution as a series expansion in terms of appropriate basis functions. The discretization yields a set of coupled nonlinear ordinary differential equations with time-dependent coefficients in the generalized coordinates which can be integrated by standard step-by-step algorithms, such as Newmark- method associated with a full (or modified) Newton-Raphson iterative procedure. So operating, however, cable deflection is represented as a continuous function, which is not able to capture the discontinuity in the slope at the time-varying mass locations. To overcome the aforementioned drawbacks and thus improve the convergence of the conventional series expansion, an alternative representation of the cable response is derived here. The main idea stems from the well-known "mode-acceleration" method, which has already been exploited by other authors [7] to improve the convergence of the conventional series expansion in the evaluation of bending moment and shear force laws along distributed parameter systems carrying one or more moving subsystems. Specifically, the improvement is achieved by adding to the conventional series expansion a correction term which explicitly accounts for the "quasi-static" contribution of the truncated high frequency modes. In order to asses the convergence of the proposed series expansion, a numerical application concerning a suspended cable under two masses travelling with a constant speed is included in the paper. It is shown that very few terms of the improved series are enough to capture the discontinuities of slope at the contact points between the cable and the moving masses. The fundamental role played by the convective acceleration terms is also demonstrated through numerical results.

1

H.M. Irvine, "Cable Structures", The MIT Press, Cambridge, 1981.

2

M. Al-Qassab, S. Nair, "Wavelet-Galerkin Method for the Free Vibrations of an Elastic Cable Carrying an Attached Mass", Journal of Sound and Vibration, 270, 191-206, 2004. doi:10.1016/S0022-460X(03)00490-5

3

L. Frýba, "Vibration of Solids and Structures Under Moving Loads", Thomas Telford, London, 1999.

4

J.S. Wu, C.C. Chen, "The Dynamic Analysis of a Suspended Cable Due to a Moving Load", International Journal for Numerical Methods in Engineering, 28, 2361-2381, 1989. doi:10.1002/nme.1620281011

5

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

6

G. Muscolino, A. Sofi, "Dynamics of a Suspended Cable Under Moving Masses", Proceedings of Fifth International Symposium on Cable Dynamics, Santa Margherita Ligure 15-18 September 2003, 109-116.

7

A.V. Pesterev, L.A. Bergman, "An Improved Series Expansion of the Solution to the Moving Oscillator Problem", Journal of Vibration and Acoustics (ASME), 122, 54-61, 2000. doi:10.1115/1.568436

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