Cable vibration reduction with a hung-on TMD system. Part I: Theoretical study

A linear theory for the free and forced vibration problems of a slightly sagging planar inclined cable with a Tuned mass damper (TMD) installed is proposed using an analytical approach in this paper. Parameters of the cable such as cable geometry-elasticity parameter, cable internal damping, and cable inclination, parameters of the damper such as damper position and damper damping ratio, and parameters of the cable-damper proportion such as the mass ratio and the stiffness ratio, are taken into consideration in the derived equations. Special cases are discussed and some simpler models can be obtained. Complex solutions of the derived equations and their physical meaning are discussed. This paper serves as a theoretical base for the companion paper on the applications of TMDs to stay cables through a parametric approach.