Stability in Parametric Resonance of a Controlled Stay Cable with Time Delay

The stability of the parametric resonance of a controlled stay cable with time delay is investigated. The in-plane nonlinear equations of motion are initially determined via the Hamilton principle. Then, utilizing the method of multiple scales, the modulation equations that govern the nonlinear dynamics are obtained. These equations are then utilized to investigate the effect of time delays on the amplitude and frequency-response behavior and, subsequently, on the stability of the parametric resonance of the controlled cable, that it is shown to depend on the excitation amplitude and the commensurability of the delayed-response frequency to the excitation frequency. The stability region of the parametric resonance is shifted, and the effects of control on the cable become worse by increasing time delay. The work plays a guiding role in the parametric design of the control system for stay cables.