An Explicit Solution of Modal-Damping Ratios for Higher Modes of a Cable with an External Damper

Design of external dampers for stay cables requires repetitive evaluation of damping ratios for target modes of cables under various damper parameters and installation positions, and explicit formulas with high accuracy to estimate the modal-damping ratios for a cable-damper system are helpful in designing external dampers. This study deals with the evaluation of the modal damping of cables with an external viscous damper close to the anchorage. A considerably accurate and explicit formula for calculating the modal-damping ratios of the cable-damper system is derived by evaluating the imaginary and real part of the complex frequency equation. It is demonstrated that the formula is able to predict the modal-damping ratios of the cable-damper system both for lower and higher modes. Based on the proposed formula, the effects of the modal order and installation position on the modal-damping ratio of the cable-damper system are studied. A closed-form solution for the optimal damping coefficient is proposed and the maximum attainable modal damping is also obtained.