Suspension Bridges with Vibrating Cables: Analytical Modeling of the Fractional-Order Resonance

This article proposes an analytical model for suspension bridges with the vibrating cables in frame of coupled systems of fractional-order boundary value problems. The proposed model includes particular multi-point boundary conditions describing interrelationship between the main body of the bridge and suspending cables. The governing problems of this analytical model are indeed coupled nonlinear differential equations of different fractional orders. The main aim of this investigation is to solve the model making use of the fixed point theory and the, stabilize it. Solvability process of this analytical model is given separately according to the resonance and non-resonance cases arising from boundary conditions. Making use of the Green function technique included in a fixed point problem, a novel uniqueness criterion is proposed to guarantee the solvability of the fractional-order model under investigation in the non-resonance case. Besides, in order to find the same solvability criteria in the resonance case, the coincidence degree theory is chosen. Next, we have a stability analysis in the sense of Mittag-Leffler for the vibrating cables of the bridge. The practical efficiency of the theoretical findings is supported by numerical prototypes.