Self-excited Oscillation And Its Optimal Control Analysis Of An Inclined Taut Cable Under Wind Loading

The fluid-flow-induced cable vibration is an active research subject in structural engineering and has practical importance for structural improvement and vibration control. This paper focused mainly on the analysis of the self-excited oscillation of an inclined taut cable with multi-vibration modes under wind loading as the cable's cross-section shape changes because of rain or snow. Firstly the nonlinear differential equations of motion were derived for the wind-induced vibration of the cable. Then the partial differential equation for the transverse cable vibration was obtained based on the assumption of longitudinal cable vibration comparatively small. By using the Galerkin approach, this partial differential equation was converted into ordinary differential equations which describe the self-excited oscillation of the cable as multi-modes system. The analytical solutions in self-excited oscillation were obtained further by using the averaging method for nonlinear vibration. According to the stability condition of analytical solutions, the wind-induced limit cycle of self-excited oscillation and its existence conditions and the critical wind velocities for the first two modes and the first, third modes of the cable system in self-oscillation were analytically determined finally and verified by the numerical results. Then the self-oscillation characteristics for the first three modes of the cable system were analyzed by numerical method. The effects of parameters about self-oscillation for the first two modes of the cable were researched at last.Secondly the cable control for the first two modes in self-excited oscillation was focused on. The performance index for optimal cable control was given to the polynomial control solution. Based on the dynamical programming principle, the Hamilton-Jacobi-Bellman (HJB) equation was established for the cable control with the index. The optimal nonlinear control force was determined from solving the HJB equation. By using the averaging method for nonlinear vibration, the analytical solutions, in particular, limit cycle solutions and stability of the controlled system in self-excited oscillation were obtained further. It was concluded by the analytical and numerical comparison between controlled and uncontrolled self-excited oscillations and stability that the proposed optimal control could effectively suppress the self-excited oscillation of the cable under wind loading. Then the control effectiveness was compared between the nonlinear optimal control and linear sub-optimal control. The results showed that under a small initial disturbance, the nonlinear optimal control and linear sub-optimal control could effectively suppress the cable oscillation and the control effectiveness was almost same; under the large initial disturbance, the nonlinear optimal control and linear sub-optimal control was also able to effectively suppress the cable oscillation but compared with the linear sub-optimal control, nonlinear optimal control could reduce the amplitude faster. Then the implementation of nonlinear optimal control was researched. It was established that one-point and two-point nonlinear optimal control of the cable in transverse direction. The results showed that selecting the appropriate control parameters, one-point and two-points nonlinear optimal control could both effectively suppress the self-excited oscillation of cable but the control effectiveness of two-point nonlinear optimal control was better than that of one-point nonlinear optimal control.Thirdly it was considered that the effects of the cable's longitudinal vibration on the cable's transverse multi-modal vibration. The differential equations of motion for the two-dimensional longitudinal and transverse coupling of cables were established. Then the self-excited oscillation characteristics for the first mode of cable longitudinal vibration and the first two modes of cable transverse vibration were analyzed by the averaging method for nonlinear vibration. The results showed that the amplitude of the longitudinal first mode was proportional to that of the transverse second mode of the cable and, the phase difference between the longitudinal first mode and the transverse second mode of the cable was 180°. Compared with those in the second chapter under the longitudinal vibration simplified, the stability of limit cycles in self-excited oscillation for the first two modes of cable transverse vibration had the same conclusion.