This paper aims to develop a dual hysteretic damper (DHD) effective for a broader class of ground motions, which includes two level-oriented hysteretic dampers and a gap element. To reveal the influence of DHD parameters on the earthquake structure response, a closed-form solution is used for the maximum response of a single-degree-of-freedom system with DHD under the critical double impulse. An energy balance approach plays a central role in the derivation of such closed-form solution. A design process of DHD in a multi-degree-of-freedom system is proposed which is based on the sensitivity analysis. The transformation of earthquake ground motions into the double impulse overcomes the difficulty in the emergence of sensitive response variation to design parameters. It is demonstrated that DHD is effective for both small and large-amplitude input motions and the proposed system is applicable to recorded ground motions approximately. It is also verified through the comparison with the designs obtained by the Monte Carlo simulation and the genetic algorithm (GA) that the proposed design method is more effective and efficient than conventional methods.