An exact analytical formulation of cable-damper system with fully dynamic boundary conditions

Cables have been extensively applied in civil engineering structures, particularly in long-span flexible bridges. Serving as critical load-bearing components, these cables inevitably operate in harsh environments, which may give rise to harmful vibrations. In practice, the boundary conditions are difficult to accurately describe owing to the dynamic contribution from the host structure, and furthermore, the consideration of dampers is challenging due to the additional lumped force. To overcome the above, an analytical cable-damper model with exact dynamic boundaries (C-D-DB model) is developed in this study, aiming to evaluate the dynamic characteristics of the cable. First, the general solution to the dynamic displacement of a cable is analytically derived. Next, a dynamic condensation technique is applied to obtain the full dynamic boundary conditions of the cable, which are characterized by frequency-dependent mass and stiffness from the host structure. Based on these, the displacement of the damped cable is formulated as a piecewise analytical solution, where the displacement function is explicitly split into multiple segments at the damper installation positions, and the governing equations for free vibration are derived from Hamilton’s principle. Such a systematic equation can be solved by the Newton-Raphson iteration, resulting in the circular frequencies and the exponentially decaying factors. Finally, the proposed methodology is verified through two numerical examples including a single cable and a suspension bridge model. By comparing with the refined finite element (FE) model, the maximum relative errors are 0.35% for the natural frequencies and 6.8% for the damping ratios. It has been shown that the proposed analytical modeling methodology has broad potential for the optimal design and health assessment of damped cables.