Transmissibilities for two-degree-of-freedom (2DoF) passive systems have been extensively investigated in the past. However, for semi-active control systems, the transmissibilities are rarely analyzed. The expressions of non-dimensional transmissibilities for the semi-active control systems are not found. There are no closed-form transfer functions for general semi-active control systems. The control algorithms for semi-active systems need to be specified first. Therefore, three reference systems (skyhook, groundhook, and their hybrid) are proposed. These equivalent systems are the references for the semi-active control systems to achieve. In this paper, the 2DoF skyhook, groundhook and hybrid systems for semi-active control are studied. The displacement and acceleration transmissibilities of those three reference systems are formulated and analyzed. Their transmissibilities are also compared with the passive system. Each of reference systems is classified as Ideal and Non-Ideal systems. The differences between the Ideal and Non-Ideal systems are also discussed. The optimal systems and their corresponding parameters are identified.
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