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Shock and Vibration
Volume 16, Issue 6, Pages 623-635

A Force Method Model for Dynamic Analysis of Flat-Sag Cable Structures

Xing Ma and John W. Butterworth

Department of Civil and Environmental Engineering, University of Auckland, Private bag 92019, Auckland 1142, New Zealand

Received 21 September 2008; Revised 20 December 2008

Copyright © 2009 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A new force method is proposed for analysing the dynamic behaviour of oscillating cables with small sags. The accepted dynamic model of such cables reduces to a partial differential equation (the equation of motion) and an integral equation (the compatibility equation). In the paper, D’Alembert’s travelling wave solution is applied to the partial differential equation (PDE). Substituting the solution into the compatibility and boundary conditions, the governing equation is obtained in terms of the dynamic tension increment. This equation has been named the force method dynamic equation (FMDE). In this way the infinite-degree-of-freedom dynamic system is effectively simplified to a system with only one unknown. Explicit solutions for both single-span and multi-span cable systems are derived. The natural frequencies obtained from the FMDE are shown to be identical to those deduced using the conventional displacement method (DM). Nonlinear governing equations are developed by considering the effect of quadratic and cubic displacement terms. Finally, two examples are presented to illustrate the accuracy of the proposed force method for single and multi-span cable systems subjected to harmonic forces.