Nonlinear parametric vibration with different orders of small parameters for stayed cables

For long-span cable-stayed bridges, parametric vibration associated with the axial component of the motion, therefore, could lead to serious dynamic instability or fatigue damage accumulations on cables, etc. Due to different stiffness and length of the cables, the governing perturbation equations of vibration for the cables vary with each other leading to different relationships of frequency and amplitude. In the present study, a theoretical model of the parametric vibration for a stayed cable subjected to the axial excitation is used, and the static sag of the cable is considered as the parabola with the only inclusion of the first mode of the cable. Governing vibration equation is obtained, consisting of nonlinear terms, namely external oscillation term, parametric oscillation term, cubic term, and quadratic term. Multiple scales method is applied to solve the governing nonlinear vibration equation. Based on the data for cables of the Shanghai Yangtze River Bridge, the coefficients of nonlinear terms are obtained, and the perturbation orders of nonlinear terms are determined. Based on the perturbation order of nonlinear terms, three types of perturbation equations motion are built based on different physical properties of cables, such as the length and cross-sectional area, etc., and the amplitude of the vibration for the deck. Therefore, the cables with different physical properties could have different parametric vibration characteristics. The results show that the sag effect of the cable has to be included when the order of deck vibration amplitude is lower than that of the parametric vibration of cable. However, the sag effect can be ignored in short cables with a large inclination angle. Meanwhile, when the external oscillation term has the same order as the quadratic term does, a combination of parametric resonance vibration and sub-harmonic occurs in the cable.