Closed-Form Formula of the Transverse Dynamic Stiffness of a Shallowly Inclined Taut Cable

Abstract

The segmented vibration-governed equations and their general solutions for cables acted upon by intermediate transverse forces are derived by applying Hamilton’s principle. Including the effects of sagging, flexible stiffness, clamped boundary conditions, and inclination angle of the cable, the element-wise dynamic stiffness for each cable segment, split into segments having unique transverse forces, is derived. By using methods from the global stiffness assembly process of FEM, the global level of the cables’ dynamic equilibrium equation is obtained, and, as a result, the final closed-form formula of transverse dynamic stiffness is derived. Additionally, the corresponding analytic form, without considering sagging effects, is also obtained. Case studies are conducted on the aspects of accuracy, rationality of the distribution on the spatial field, and frequency domains of dynamic stiffness calculations. By comparison with the Guyan-based static FEM reduction method, it is shown that the result obtained from the proposed closed-form solution, which includes sagging effects, is exact and rational, thus creating a powerful tool in transverse vibration analysis.