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Summary Inerter dampers (IDs) and other inerter‐based vibration absorbers have elicited growing interest for vibration mitigation of stay cables. This study systematically investigates the vibration mitigation mechanism of tuned IDs (TIDs) for stay cables based on a continuous cable model. A TID consists of an ID connected with a spring stiffness in series (TID‐S) or parallel (TID‐P). On the basis of systematical parametric analyses, the impact of stiffness in series or parallel connections is evaluated, and the interrelations among ID, TID‐S, and TID‐P are elaborated. Subsequently, a detailed tuning procedure is summarized. Two optimal tuning formulas are also obtained to facilitate the rapid design of TID‐S and TID‐P. According to the optimized results, the damping ratios contributed by ID, TID‐S, or TID‐P to a stay cable are essentially determined by the inertance value. When the inertance deviates from the optimal value, the performance of ID drops significantly, but this adverse effect can be mitigated by tuning the stiffness in TID‐S or TID‐P.

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