Semi-active control of shallow cables with magnetorheological dampers under harmonic axial support motion

The paper deals with the control of sub- and superharmonic resonances by means of magnetorheological (MR) dampers of an inclined shallow cable caused by parametric excitation from harmonically varying support points. A mechanical model based on the Dahl hysteretic model is used to describe the dynamic property of the MR damper, and the finite difference method is used to discretize the derived governing nonlinear equations of motion of the cable–damper system. Semi-active control strategies are proposed based on the modulated homogeneous friction (SA-1) algorithm and the balance logic (SA-2) algorithm. Four cases are analysed when the circular frequency Ω of the support point motion is in the vicinity of 2ω1, ω1, 2ω1/3, and ω1/2, ω1 being the first in-plane eigenfrequency of the cable. The vibration reduction ability of the MR damper is compared with that of the viscous damper optimally tuned to the first in-plane eigenmode of the cable. The numerical results show that the MR damper with the SA-1 rule and the optimal viscous damper perform similarly to mitigate the vibration of the cable under axial periodic support motion, and, in some cases, the SA-2 rule is more favourable in suppressing the cable vibration compared with the SA-1 rule. In the final analysis, both the MR damper and the viscous damper can effectively mitigate the out-of-plane component of the cable, while having little effect on the reduction of the in-plane response in most cases. Furthermore, when the support motion amplitude is located in a certain range for the case Ω≈2ω1/3, the original zero out-of-plane vibration of the cable should be changed to the stable in-plane and out-of-plane coupled oscillation by using the optimal passive viscous damper or the MR damper with the SA-1 rule. It is also observed that the vibrations of the cable become chaotic when the cable is supplied with the MR damper using the proposed nonlinear control algorithms.