Free vibrations of a taut cable with a general viscoelastic damper modeled by fractional derivatives

This study extends dynamic understanding of a taut cable with a viscous damper at arbitrary location to that with a general linear viscoelastic (VE) damper portrayed by a five-parameter fractional derivative model (FDM). The FDM is able to describe a generalized relationship between force and deformation of viscoelastic dampers (material) in a wide frequency range, which can simulate a practical damper including its support condition or a secondary tie between neighboring cables. Free vibrations of the passively controlled cable system have then been formulated analytically through complex modal analysis. For the restricted case that the FDM is installed close to one cable anchorage, asymptotic solutions for the system complex frequency and modal damping are presented; explicit formulas have also been derived to determine the maximal attainable damping and corresponding optimum FDM parameters, based on which effects of frequency-dependent damper properties are appreciated. Considering the FDM located at arbitrary location, the three distinct regimes of frequency evolutions observed for a cable with a viscous damper have been generalized to that with a VE damper; also, new characteristics of the regime diagram and the frequency evolution in each regime are observed.