Parametrically excited instability analysis of a semi-actively controlled cable

A semi-active optimal control method for the parametrically excited instability of stay cables is developed based on the LQR control and the Bingham model of MR dampers, and the instability of controlled inclined stay cables under support motions is studied through the cable structure and control factor effects. The partial differential equation of the parametrically excited transverse vibration of a controlled inclined stay cable with sag under support motions is derived by using two transformations of displacements and converted into ordinary differential equations for the cable mode vibration according to the Galerkin method. The optimal control law is obtained by applying the linear quadratic regulation control method to the multi-degree-of-freedom cable system and matched with the MR damper dynamics to determine the semi-active optimal control force. Then the unstable regions of the parametrically excited vibration of the controlled cable are calculated by using the direct numerical approach based on the generalized eigenvalue analysis and revised through numerical simulation to illustrate more effective stability or large-amplitude vibration suppression. The influences of the cable structure parameters and control factors on the parametrically excited instability are analyzed based on large numerical results in terms of unstable regions and minimum parameter-excitation amplitude. The parametrically excited instability of the cable vibration is noticeably affected by decreasing the inclination angle close to the support-motion direction and increasing the time delay in control over a certain value, which effects need to be taken into account for the cable instability control.