Stability analysis for active control with a sky-hook and ground-hook inerter-damper configuration

Abstract—In this paper, a sky-hook and ground-hook inerter-damper configuration is proposed, and its active realization using active control actuators is studied as it cannot be directly implemented in practice. Especially, the stability problem caused by the active realization is analyzed. A general two-degree-of-freedom model which can be used to model various mechanical systems including quarter-car models and dynamic vibration absorbers is employed to analyze the influence of the passive inerter and the skyhook/ground-hook inerter on the stability of the overall active control system. It is demonstrated that for the sky-hook damper control, the passive inerter has negative effect on the stability; while for the ground-hook damper control, the passive inerter has positive effect on the stability. In contrast, the sky-hook inerter has positive effect on the stability and the ground-hook inerter has negative effect on the stability. The influence of the passive inerter and the skyhook/ground-hook inerter on the sky-hook and ground-hook damping coefficient is also investigated, where it is shown that the passive inerter decreases the admissible range for the sky-hook damping coefficient, and increases the admissible range for the ground-hook damping coefficient. Meanwhile, the sky-hook inerter increases the admissible range for the sky-hook/ground-hook damping coefficient, while the ground-hook inerter decreases the admissible range for the skyhook/ground-hook damping coefficient.