Parameter analyses of suspended cables subjected to simultaneous combination, super and sub-harmonic excitations

The present study is dealing with a geometrically nonlinear model of the suspended cable subjected to multi-frequency excitations. In particular, two of the super, sub-harmonic, and combination resonances are excited simultaneously here. The nonlinear integro-differential equations of the suspended cable are introduced, together with the Galerkin method, to obtain a reduced-order model, whose responses are investigated by solving the reduced ordinary differential equations. Then, the obtained single-mode discretization equations are solved using the method of multiple scales in the frequency regions of three simultaneous resonant cases, with the stability characteristics determined. Effects of parameters on resonance characteristics are carried out by investigating several numerical examples. Numerical results demonstrate that the two-frequency excitation has significant influences on the dynamical behaviors of the nonlinear system. Each harmonic excitation component's contribution to the overall resonant responses is mainly dependent on its excitation amplitude. The stable steady-state solutions are confirmed by using numerical integration, and the number of steady-state solutions varies from one to seven as to different parameters of the system in simultaneous resonances. Besides, it is of great significance to include the effects of excitation phase differences on nonlinear vibration characteristics.