Nonlinear vibration of the cable-stayed shallow arch subjected to a repeatedly moving harmonic load: dynamic modeling and one-to-one internal resonance

To investigate the effects of moving vehicles on the dynamic behavior of long-span cable-stayed bridges, this study establishes a cable-stayed shallow arch model by considering both cable vibration and the initial configuration of the bridge beam. The moving vehicle is modeled as a repeatedly and uniformly moving harmonic force, and the dynamic response characteristics are analyzed when the global and local modes of the structure exhibit a one-to-one internal resonance relationship. First, the governing equations of the system are derived based on Hamilton's principle. Subsequently, the resulting partial differential equations are discretized into a set of ordinary differential equations (ODEs) using the Galerkin method. The obtained ODEs are then solved via the method of multiple time scales (MMS). On this basis, simultaneous resonance (primary resonance and subharmonic resonance) caused by the repeatedly moving harmonic force is studied. The frequency-response curves and force-response curves of the system are provided to examine the influence of key parameters on the nonlinear dynamic behaviors of the structure. Simultaneously, two-parameter bifurcation diagram, the time-histories, phase portraits and power spectrum are analyzed. The results show that the required excitation amplitude differs for distinct saddle-node bifurcations. Moreover, subharmonic resonance may not be dominant unless the excitation amplitude is relatively large.