The response phase of a nonlinear vibrating cable under fixed excitation frequency is commonly believed to be a constant value in the linear solution, while the higher-order terms (HOTs) effect is omitted. However, as the variation of cable parameters, the HOTs would have a significant contribution to the response and thus changes the phase instantaneously. In order to ascertain the instantaneous phase difference between cables with the consideration of the HOTs, the instantaneous phase-frequency characteristics of two suspended cables with different parameters are investigated. The dimensionless dynamic equations of a two-cable system were derived, and the discrete ordinary differential equations were obtained using the Galerkin method and then solved by the Method of Multiple Scales (MSM). The MSM solution was verified simultaneously using the Runge-Kutta method (R-K) and the Finite Element Method (FEM). The numerical analysis shows that the HOTs' influence on the instantaneous phase cannot be ignored in some frequency range. The origination of the instantaneous phase difference between the two cables with different parameters comes from two aspects: (i) the difference in the phase shift values (PSVs) in the linear term; and (ii) the difference in the proportion of the drift terms (DTs) in the HOTs.

No abstract

The response phase of a nonlinear vibrating cable under fixed excitation frequency is commonly believed to be a constant value in the linear solution, while the higher-order terms (HOTs) effect is omitted. However, as the variation of cable parameters, the HOTs would have a significant contribution to the response and thus changes the phase instantaneously. In order to ascertain the instantaneous phase difference between cables with the consideration of the HOTs, the instantaneous phase-frequency characteristics of two suspended cables with different parameters are investigated. The dimensionless dynamic equations of a two-cable system were derived, and the discrete ordinary differential equations were obtained using the Galerkin method and then solved by the Method of Multiple Scales (MSM). The MSM solution was verified simultaneously using the Runge-Kutta method (R-K) and the Finite Element Method (FEM). The numerical analysis shows that the HOTs’ influence on the instantaneous phase cannot be ignored in some frequency range. The origination of the instantaneous phase difference between the two cables with different parameters comes from two aspects: (i) the difference in the phase shift values (PSVs) in the linear term; and (ii) the difference in the proportion of the drift terms (DTs) in the HOTs.