About one half of the 25 billion square meters of built-up area in Europe were constructed before 1960 and are highly vulnerable to earthquakes. Conventional seismic upgrading techniques can prevent collapse in the event of a strong earthquake but cannot avoid severe damage to the existing structure and the need for demolition and reconstruction. In contrast, innovative seismic upgrading techniques using energy dissipation devices (dampers) can minimize the damage and avoid demolitions and heavy reparations. This paper presents an ongoing research project aimed at developing new advanced damper that depends on both the displacement and the velocity and harness the dynamics of multistability through tailored metamaterials. To date, the energy-balance based approach has fallen short of its true potential and there are many aspects that need research: particularly its application to structures with advanced dampers dependent on both displacement and velocity. This paper presents the idea of the new multistable damper and discusses how the fundamental energy-balance equation can be specialized for structures equipped with the new device, to produce reliable and simple formulations that can be used in practical design.