**4. Analytical Models of Dampers**

Four types of dampers, i.e., tuned mass dampers (TMDs), tuned liquid dampers (TLDs), particle dampers (PDs), and particle-tuned mass dampers (PTMDs), are selected to mitigate the seismic responses of the four-story steel frame. All four types of damper are installed on the top story and can be regarded as secondary structures to the primary structure, so they are ideal experimental substructures for SSTT. The four structure–damper systems are illustrated in Figure 6. The TMD can absorb the vibration energy at the tuned frequency and, therefore, reduce the structural response of the primary structure. The TLD is a tank containing liquid and can dissipate the vibration energy by liquid boundary layer friction, free surface contamination, and wave breaking. Similarly, the PD is a container equipped with particles that dissipate the energy by particle-to-particle and particle-tocontainer wall collision and friction. The PTMD is a combination of a TMD and PD, so it integrates the energy dissipation mechanisms of the two dampers.

**Figure 6.** Primary structure–secondary structure (damper) systems.
