5.3.2. Earthquake

A series of RTHSs were conducted on a spring specimen with a stiffness of 35 kN/m. Three different numerical substructures, whose stiffnesses were 35, 17.5, and 0 kN/m, were considered. The damping ratio was 0.1. For the NS with a stiffness of 35 kN/m, the period was 2 s. The earthquake excitation was El Centro (NS, 1940), and the peak ground acceleration (PGA) was scaled to 50 gal. The displacement time histories, overall and enlarged view, are shown in Figure 23, together with the reference solution.

**Figure 23.** Displacement time histories under earthquake excitation. (**a**) Overall view for Case 1; (**b**) enlarged view for Case 1; (**c**) overall view for Case 2; (**d**) enlarged view for Case 2; (**e**) overall view for Case 3; (**f**) enlarged view for Case 3.

It is seen in the figure that the measured displacements were almost identical to the command for the three cases, indicating a perfect tracking performance of the *H*∞- controlled system. They were further verified by the RMSE collected in Table 3.


**Table 3.** Stiffness of NS and RMS error of hybrid simulation.

If focusing on the displacement response and reference solution, one will find that the displacements of RTHS matched the reference well before the first displacement peak. Subsequently, differences emerged, especially at each positive or negative peak.

It is seen in the figure that the tracking performance was perfect, but the simulation results were barely satisfactory. The reason is that the measured force was not synchronized with the desired displacement. There were two different filters, which may introduce additional time delays differently.
