Dynamic analysis of cables with variable flexural rigidity

Abstract

In general the governing equation for cable dynamics is a nonlinear partial differential equation with the fourth-order derivative in the space domain and the second-order derivative in the time domain. Analytical solutions are impossible by current available methods, and numerical methods are needed. In this research, several approaches are examined to simplify this type of nonlinear equation. The nonlinear properties of cables are mostly caused by internal damping. According to experiments[3] at least two sources of internal losses can be identified in a vibrating cable: (1) those due to dry friction between the wires; and (2) those due to viscous damping of the material of which the wires are made [9]. The former is amplitude-dependent but frequency-independent, whereas the latter is amplitude-independent and frequency dependent. The combination of these two mechanisms of losses results in hysteretic loop shapes, which are amplitude and frequency dependent. In this study, viscous damping is neglected in comparison with the large dry friction effects. Since dry friction is frequency-independent, its effect can be studied on the basis of a static approach. As is well known, an undamped cable can be modeled as a BERNOULLI-EULER beam under strong tension, with the bending moment at each point is proportional to the local curvature. But for cables where internal friction exists, the relation between an instantaneous bending moment and curvature is no longer linear but is described by hysteresis loops. The CableCAD®' [lO] software can predict the moment curvature relationship (hysteretic loop shape) based on a static approach. By using this relationship, the value of moment at any point along the deflected cable can be calculated. In this study, the cable is modeled as a continuous beam, and instead of adding extra damping term to the governing equation, only the moment-curvature relationship predicted by the CableCAD® software is used to calculate the value of flexural rigidity and bending moment. This frictional bending model (FBM) appears to be a reasonable approach because there are no external dampers attached to vibrating cables. In this approach, the fourth-order derivative partial differential equation can be reduced to second-order derivative partial differential equation, and solved by the finite difference method (FDM).