An asymptotic expansion of cable–flexible support coupled nonlinear vibrations using boundary modulations

This paper is devoted to cable–flexible support coupled nonlinear vibrations using a asymptotic boundary modulation technique. Asymptotic/reduced cable–support coupled nonlinear models are established first using the boundary modulation concept, after a proper scaling analysis at the cable–support interface. The cable and the support turn out to be coupled through cable-induced and support-induced boundary modulations in a rational way, which are derived analytically by asymptotic approximations and multiple scale expansions. Based upon the reduced models, two prototypical kinds of cable–support coupled dynamics are fully investigated, i.e., one with the support excited and the other with the cable excited. Essentially, they correspond to refined versions of two typical degenerate cable dynamics, i.e., cables excited externally with fixed supports and cables excited by ideal moving supports. Applying numerical continuation algorithms to the reduced models, cable–support typical coupled frequency response diagrams are constructed, with their stabilities, bifurcation characteristics, and the coupling’s effects on the cable determined. All these approximate analytical results are verified by the numerical results from the original full cable–support system using the finite difference method.