Achieving upper-branch steady-state solutions for inerter-damped cable parametric vibration via targeted runge-kutta initial conditions.

Increasing the span length of stay cables markedly increases their susceptibility to parametric resonance. This study investigates mitigating parametric resonance in stay cables using a parallel inerter damper and focuses on a methodology to obtain the upper-branch steady-state solution via Runge-Kutta integration. The approach computes both transient and steady-state responses under arbitrary support excitation. Furthermore, we examine the dependence of the Runge-Kutta steady-state solution on initial conditions by analyzing phase portraits of the cable-inerter system. Based on the method of multiple scales, we propose a general procedure to select initial conditions that ensure convergence to the upper-branch steady-state solution. The results show that different initial conditions drive the transient Amplitude along different phase trajectories and may converge to distinct steady-state solutions. The inerter damper modifies the critical phase trajectory of the cable-damper system, thereby shifting the feasible region of initial conditions. Thus, when initial conditions from the undamped case are used in the presence of an inerter damper, the Runge-Kutta solution may converge to the lower-branch stable solution, which may overestimate the damper's control effectiveness.