To reduce large displacement demand of traditional base-isolation (TBI) system for building structures and address the problem of equivalent simulation method of inerter, this paper proposes a novel accelerated oscillator damper (AOD) base-isolation system and the displacement consistentequation method. The AOD consists of transmission device (rack and pinion mechanism), auxiliary mass, spring, and dashpot. The transmission device amplifies the relative motion of the buildings to ground and transfers to the auxiliary spring, dashpot and mass so that enhancing the capacity of energy absorption. The H2 norm criterion is introduced to achieve optimum mean square (MS) vibration performance of AOD-BI system. The AOD frequency ratio and damping ratio are selected as the design parameters, whose closed-form solutions are obtained for the undamped primary structure and numerically verified. The influence of the structural parameters on the design parameters is also studied. Optimum MS vibration performance comparisons are conducted for the traditional, optimum AOD, tuned mass damper (TMD) and serial tuned inerter damper (TID) base-isolation systems both in frequency domain and in time domain. The results show that the optimum AOD-BI system is effective in suppressing the base drift, acceleration of superstructure and relative displacement between the base floor and superstructure with less stroke of the auxiliary mass. In addition, a convenient method, the displacement consistentequation method for modelling an inerter damper is studied through a conventional finite element (FE) commercial software.