Study on nonlinear behaviors of a double-cable-stayed beam model with various trial functions of the cables

To explore the internal resonance between different cables and beam in the cable-stayed bridge, a double-cable-stayed beam model is established in this paper. Unlike previous studies, the two cables have different frequencies. Therefore, it is possible to investigate the in-plane one-to-one-to-two internal resonance among the second-order mode of the beam (global mode) and the first-order modes of two cables (local modes) when the beam is subjected to the primary resonance excitation. Firstly, the partial differential equations (PDEs) governing the behaviors of the beam and cables are discretized using the Galerkin method, resulting in ordinary differential equations (ODEs). It is worth noting that two various trial functions of the cables, namely, the superposition of dragging function with sinusoidal function (TF1) and real modal function (TF2), are adopted to compare the differences in nonlinear behaviors, which is also the motivation of this study. Then, the multiple time scale method is employed to solve the ODEs, and in this way, the corresponding modulation equations are derived. Based on the obtained modulation equations, the steady-state solutions are acquired so as to discuss the nonlinear characteristics of the system using various trial functions. Meanwhile, the accuracy of the perturbation results is verified by solving the ordinary differential equations directly with the Runge–Kutta method. Finally, several interesting findings and conclusions are given.