*4.2. Virtual RTHS*

In this section, virtual RTHSs were conducted on a single-degree-of-freedom (SDOF) structure. The mass was 1200.8 kg, the structural stiffness was 2276.3 kN/m, and the damping ratio was 5%. A nonlinear PS was employed, of which the mass and stiffness of the PS were 56.6384 kg and 2276.3 kN/m, respectively, the yield displacement was 7.6 mm, and the stiffness coefficient after yield was 0.12. The earthquake excitation was El Centro (NS, 1940), whose peak ground acceleration (PGA) was scaled to 0.16g. In the virtual hybrid simulation, the *α* method [26] was adopted to solve the equation of motion, and the time integral was 0.01 s. A pure dynamic analysis of the whole structure was performed to serve as an exact reference solution, where the stepwise integral method was CDM-RST [25], and the time step was 0.01 s. Due to the PS being changed, the controller was re-designed, and the controller was:

$$K = \frac{14,893,750.8909(s + 2.094 \times 10^4)(s + 1.047 \times 10^4)(s + 1.413)(s^2 + 164.3s + 3.704 \times 10^5)}{(s + 1.305 \times 10^6)\ (s + 2.096 \times 10^4)\ (s + 1.031 \times 10^4)\ (s + 1196)\ (s + 669.9)\ (s + 0.001004)} \tag{17}$$

The time histories of displacement obtained by virtual RTHS were compared with those of the reference solution, which is shown in Figure 17. It is seen in the figure that a shifting was observed for the displacement due to the yielding of the structure. However, the responses were in good agreemen<sup>t</sup> with those of the reference solution under the earthquake excitation.

**Figure 17.** Displacement time histories for virtual RTHS.
