Abstract: An inerter system can absorb vibration energy efficiently and amplify the deformation of its internal energy dissipating device (EDD), thereby improving the EDD’s efficiency of energy dissipation and response mitigation. This is called the characteristic of damping enhancement, which is the key mechanism of the inerter system and can be used as a concise design principle for the inerter system. However, the existing damping-enhancement formulae for the design of inerter systems are complicated and inconvenient to apply. To give full play to the damping enhancement characteristic of the inerter system, the “maximum damping enhancement” is adopted as the design principle, and the satisfaction of the performance demands is taken as the direct goal. And then the closed-form design formulae are obtained. Based on the theory of random vibration, the closed-form formulae of the mean-square responses for a single-degree-of-freedom (SDOF) structure with an inerter system are obtained under the white noise excitation, and the expressions of optimal parameters of the inerter system are derived according to the extremum conditions and the non-inherent-damping assumption. The design process of the inerter damping system is provided based on the principle of damping enhancement and the closed-form formulae. The corresponding computer program is developed for the design and performance verification of SDOF structures with inerter systems. The analysis results of the design cases confirm the correctness and validity of the design principles and the simple closed-form formulae in this paper.