Nonlinear responses of suspended cable under phase-differed multiple support excitations

Cable structures may be subjected to multiple support excitations and exhibit complex nonlinear dynamic characteristics. To study the influence of phase-differed support excitations on the responses of suspended cable, a model with both ends subjected to 3-D non-uniform excitations is established, and the infinite-dimensional discrete motion equations are obtained. The in-plane primary resonance and out-of-plane primary resonance are studied using the multiple scales method. The general expressions of the equivalent excitation amplitude and the response phase of the multi-excitation system are obtained. The effects of different combinations of horizontal, vertical, and transverse excitations on the response amplitude and phase are analyzed theoretically, and the multi-scale solutions are verified by numerical integration. The results show that the multi-excited cable can be regarded as a nominal single excitation system (NSES). The effect of multiple support excitations on the cable response is mainly reflected in two aspects: First, these excitations make the response phase of the system shift based on the response phase of the NSES; second, these support excitations affect the equivalent excitation amplitude of the NSES. Both the response phase’s shift and the equivalent excitation amplitude depend on the combination of the excitations and the parity of the excited modes. When the phase difference between the two excitations is gradually changed from 0 to $$\pi $$ , the phase–frequency curve is also shifted by $$\pi $$ . However, the shifting process is not necessarily linear. The excitation phase difference does not change the soft and hard properties of the system. However, it can simultaneously change the response amplitude and the resonance region’s width so that the amplitude–frequency curve changes with a period of $$2\pi $$ in a specific region.