On Internal Resonance Analysis of a Double-Cable-Stayed Shallow-Arch Model with Elastic Supports at Both Ends

In previous research on the nonlinear dynamics of cable-stayed bridges, boundary conditions were not properly modeled in the modeling. In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately, a double-cable-stayed shallow-arch model with elastic supports at both ends and the initial configuration of bridge deck included in the modeling is developed in this study. The in-plane eigenvalue problems of the model are solved by dividing the shallow arch (SA) into three partitions according to the number of cables and the piecewise functions are taken as trial functions of the SA. Then, the in-plane one-to-one-to-one internal resonance among the global mode and the local modes (two cables\u27 modes) is investigated when external primary resonance occurs. The ordinary differential equations (ODEs) are obtained by Galerkin\u27s method and solved by the method of multiple time scales. The stable equilibrium solutions of modulation equations are obtained by using the Newton-Raphson method. In addition, the frequency-/force-response curves under different vertical stiffness are provided to study the nonlinear dynamic behaviors of the elastically supported model. To validate the theoretical analyses, the Runge-Kutta method is applied to obtain the numerical solutions. Finally, some interesting conclusions are drawn