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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 72
A Method for the Dynamic Analysis of Suspended Cables Carrying Moving Masses G. Muscolino^{1} and A. Sofi^{2}
^{1}Department of Civil Engineering, University of Messina, Italy
G. Muscolino, A. Sofi, "A Method for the Dynamic Analysis of Suspended Cables Carrying Moving Masses", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 72, 2006. doi:10.4203/ccp.83.72
Keywords: suspended cable, moving masses, convective acceleration, series expansion, modeacceleration method, quasistatic solution, slope discontinuity.
Summary
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 wellknown, 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 inplane vibrations of suspended cables with small sagtospan ratios carrying an arbitrary number of moving masses. The cable is modelled as a monodimensional 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 inplane vibrations of the cablemass 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 integrodifferential 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 timedependent coefficients in the generalized coordinates which can be integrated by standard stepbystep algorithms, such as Newmark method associated with a full (or modified) NewtonRaphson 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 timevarying 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 wellknown "modeacceleration" 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 "quasistatic" 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. References
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