Nonlinear planar vibrations of a cable with a linear damper

This paper minutely studies the effects of the damper on nonlinear behaviors of a cable–damper system. By modeling the damper as a combination of a viscous damper and a linear spring, the primary resonance and subharmonic resonance (1/2 order and 1/3 order) of the cable are explored. The equation of motion of the cable is treated by using Galerkin’s method, and the ordinary differential equation (ODE) is obtained subsequently. To solve the ODE, the method of multiple timescales is applied, and the modulation equations corresponding to different types of resonance are derived. Then, the frequency–/force–response curves are acquired by utilizing Newton–Raphson method and pseudo-arclength algorithm, so as to explore the vibration suppression effect from a nonlinear point of view. Meanwhile, the time histories, phase portraits and Poincaré sections are also provided to discuss the influences of the damping and spring stiffness on the nonlinear characteristics of the cable. The results show that the damper has a significant effect on nonlinear resonances of the cable, and the increase in damping (spring stiffness) makes the dual characteristic gradually convert to the softening (hardening) characteristic.