Study on nonlinear coupled vibrations of damaged suspended cables with symmetry-breaking

Symmetry is one of the five aesthetic characteristics in the vibration theory, but the symmetry-breaking is also inevitable. This paper takes a common vulnerable structure in engineering-the suspended cable-as an example, and the influences of symmetry-breaking on the planar coupled vibrations have been investigated when the asymmetric damage is occurred. Firstly, the in-plane nonlinear dynamical model of damaged suspended cable has been established, and the nonlinear infinite dimensional differential equations have been obtained by using the Galerkin method. The method of multiple scales has been adopted to obtain the modulation equations of the nonlinear systems' in-plane coupled vibrations. The resonant curves of undamaged and damaged suspended cables including the first nine modes have been obtained by using the numerical methods, and the stabilities of solutions have also been determined. The largest Lyapunov exponent has been calculated to determine the system's chaotic motions. The numerical results show that the classical parabolic curves have been often adopted to simulate the suspended cables' static configurations. However, when the asymmetric damage occurs, the piecewise functions should be used to accurately describe the damaged cables' static configurations. The symmetry-breaking causes crossover points between two natural frequencies of suspended cables to turn into veering points, and the symmetric/anti-symmetric mode shapes before damage are changed into the asymmetric ones after damaged. The nonlinear interaction coefficients are changed significantly, resulting in significant changes in internal resonant responses. When the excitation is directly applied to the higher-order modes, the single-mode solutions and internal resonant ones are obvious in the undamaged system, while the damaged system does not present the obvious single-mode solutions. The bifurcations and chaos of the damaged system are also changed obviously, and some chaotic motions around the period-doubling bifurcation are observed as to the damaged system. Copyright © 2022 Chinese Journal of Theoretical and Applied Mechanics. All rights reserved.