Modelling and energy transfer in the coupled nonlinear response of a 1:1 internally resonant cable system with a tuned mass damper

Abstract

Considering the vibration of a Tuned Mass Damper (TMD), a dynamic model composed of the cable and TMD is investigated. Different from the other studies, this paper is mainly devoted to nonlinear behaviours of the model by considering the participation of the damper in energy transfer and coupling interaction between the cable and damper. According to the extended Hamilton’s principle, the classical equations of motion of the cable and TMD are derived. Based on the equations of motion of the cable and TMD, the one-to-one internal resonance of the system is studied when external primary resonance of the cable occurs. By applying the Galerkin’s method, a set of ordinary differential equations (ODEs) are obtained. To solve the ODEs, the multiple time scale method is used and the modulation equations are derived. The stable solutions of the modulation equations are acquired by Newton-Raphson method and continued by pseudo arclength algorithm. Meanwhile, the parametric analyses of some key parameters, such as the excitation amplitude, the spring stiffness, the damping ratio and position of the TMD and the sag of the cable, are carried out through frequency-/force-response curves to explore the nonlinear behaviours of the system. The results show that the TMD plays an important role in both energy consumption and energy transfer.