Resonance regions due to interaction of forced and parametric vibration of a parabolic cable

Abstract

Taut cables such as those used in cable-stayed bridges are prone to exert a large amplitude response due to support motion. Support motion generally involve longitudinal and transverse components with respect to the cord. Such motions induce parametric and external excitation of the cable. For a certain excitation frequency, multimodal interaction due to simultaneous parametric and primary resonance can be expected. Previous studies have focused on changes in the boundary curves of the primary region of the parametric resonance affected by forced vibrations due to primary resonance. The hysteresis regions have not been determined; they have only been analyzed using the frequency-amplitude curves of the response. Moreover, the influence of the longitudinal and transverse displacement ratio has not been discussed thus far. This paper presents a complete formulation of the continuum equations of a cable model that is excited by support motion. A nonlinear discretized model that includes quadratic and cubic nonlinearities is obtained using the Galerkin method. Further, the analytical solution is determined using the method of multiple scales (MMS). By obtaining the expressions for the amplitudes and phases, the mathematical conditions are set for the amplitude of the parametric and primary resonance from which analytical expressions for the boundary curves of the interaction resonance region are derived. Local stability analysis is conducted for the steady-state response, and direct numerical integration is used for validating the frequency-amplitude curves obtained using the MMS. The solutions are verified using two independent numerical models. It is shown that the ratio of the transverse and longitudinal components of support motion significantly affects the resonance region, and several different response solutions can be obtained. It is also shown that different values of the mechanical cable parameters affect the resonance region and vibration amplitude.