Free Vibrations of Taut Cable with Attached Damper. I: Linear Viscous Damper

Abstract

Free vibrations of a taut cable with an attached linear viscous damper are investigated in detail using an analytical formulation of the complex eigenvalue problem. This problem is of considerable practical interest in the context of stay-cable vibration suppression in bridges. An expression for the eigenvalues is derived that is independent of the damper coefficient, giving the range of attainable modal damping ratios and corresponding oscillation frequencies in every mode for a given damper location without approximation. This formulation reveals the importance of damper-induced frequency shifts in characterizing the response of the system. New regimes of behavior are observed when these frequency shifts are large, as is the case in higher modes and for damper locations further from the end of the cable. For a damper located sufficiently near the antinode in a given mode, a regime of solutions is identified for which the damping approaches critical as the damper coefficient approaches a critical value. A regime is developed to indicate the type of behavior in each mode for any given damper location.

Performance Analysis of Cables with Attached Tuned-Inerter-Dampers

“…For example, if the damper capacity is set to ξ V D = 7.9, the corresponding modal damping ratios will be ζ 1 = 1%, ζ 2 = 0.8% and ζ 3 = 0.6%. Similar results have been reported in [4] for a different scaling of the damping parameter.…”

“…The approach is similar to the one presented in [4]. As explained by the authors, there is an optimal damping capacity that ensures a maximum level of modal damping in each vibration mode.…”

“…where λ is the dimensionless eigenvalue and γ is the displacement at the location where the device is connected [4]. The equilibrium equation at the TID connection point can be written as…”

“…Later, the universal curve formulation was extended by Cremona [3] to inclined cables. The work of Main and Jones [4] extended these studies to the case when dampers trigger large frequency shifts of the uncontrolled cable natural frequency. On the experimental side, the behaviour of cables with attached dampers was presented in several case studies [5].…”

“…Based on the analysis of horizontal cables with attached viscous dampers [4] presented in the literature, we study the performance of cables with attached TIDs. Starting from the uncontrolled cable model, differential equations can be derived for calculating the modal damping ratio obtained using a TID.…”