One-to-one internal resonance of a cable-beam structure subjected to a concentrated load

• Cable-cantilever beam with a concentrated excitation is established. • One-to-one modal resonance is investigated. • Frequency island phenomenon and complex jumping phenomena are found. • Cable has rich behaviors and significant jump phenomena. • Inverse trend of responses of the beam and the cable is discovered. • Frequency responses of the beam and cable exhibit hardening spring properties. Cable-beam composite structure, such as a cable-stayed bridge in cantilever construction and a tower crane hoisting load, will sustain dynamic excitation. In order to investigate the dynamic behaviors at this stage, a cable-supported cantilever beam model subjected to a concentrated harmonic excitation at the cantilever end of the beam is established. The interactions including the second-order effect of the axial compressive load on the beam and the dynamic interaction between the beam and the cable due to cable's dynamic elongation are considered. A reduced two-degree-of-freedom system is obtained by using Galerkin discretization. Multiple time scales method is applied to solve the ordinary differential equations. Based on the derived modulation equations and the ordinary differential equations, frequency response, amplitude response, phase plane and time history curves are provided to explore the dynamic behaviors of the system. Three different lengths of the cable-stayed cantilever beam system are considered to study the changes in dynamic behaviors. In order to verify the validity in each case, Runge-Kutta method is used to directly solve the original ordinary differential equations and two results have a satisfying consistence. The analysis results show that frequency responses of both the beam and the cable exhibit hardening spring properties; the cable has more rich and complicated dynamic behaviors than the beam, for example the cable has more significant jump phenomenon.