*5.2. Numerical Results and Discussions*

It can be seen from Figures 8 and 9 that the SSTT system is composed of four parts: the experimental substructure (damper), the analytical substructure (frame), the integration algorithm, and the shake table. Therefore, all four components influence the results of the SSTT system. The four-story frame in Section 3.3 is taken as the analytical substructure and remains unchanged. The four types of damper in Section 4 with different auxiliary mass ratios are taken as the experimental substructures. Therefore, the effects of the auxiliary mass ratio are investigated first. As discussed in Section 2, the integration parameters *κ*1 and *κ*2 greatly influence the numerical properties of the GCR algorithms. Thus, the effects of different sets of integration parameters are considered. Similarly, the time step is a critical factor determining the accuracy of the integration algorithm, and thus, is also studied. Finally, the time delay, which is a critical property of the shake table dynamics, is also taken into account.

### 5.2.1. Effects of the Auxiliary Mass Ratio

GCR algorithms with *κ*1 = 1/2, *κ*2 = 1/4 and a time step of 0.001 s are adopted to solve the virtual SSTT of the four-story steel frame (Section 3) with four types of damper (Section 4). Figure 10 shows the lateral displacements at the fourth story for the steel frame, in addition to four dampers with four auxiliary mass ratios. The seismic responses of the uncontrolled structure without dampers are used for comparison. The corresponding damper forces are provided in Figure 11. Two widely used reduction factors based on the maximum and root-mean-square (RMS) structural responses are used to quantify the reduction effects of dampers:

$$R\_{\text{max}} = 1 - \frac{\max |\boldsymbol{x}\_{controlled}|}{\max |\boldsymbol{x}\_{uncontrol}|} \tag{34}$$

$$R\_{\rm RMS} = 1 - \frac{\rm RMS(x\_{controlled})}{\rm RMS(x\_{unconrolled})} \tag{35}$$

where *xcontrolled* and *xuncontrolled* are the structural responses of the controlled structure with dampers and the uncontrolled structure without dampers, respectively. The two reduction effects of the different dampers with varying mass ratios are compared in Figure 12.

**Figure 10.** Comparisons of lateral displacements with different auxiliary mass ratios. (**a**) TMD; (**b**) TLD; (**c**) PD; (**d**) PTMD.

**Figure 11.** Time–history curves of damper forces. (**a**) TMD; (**b**) TLD; (**c**) PD; (**d**) PTMD.

**Figure 12.** Reduction factors with respect to auxiliary mass ratios for four types of dampers. (**a**) *R*max; (**b**) *R*RMS.

Figures 10 and 12 show that the TLD has the best performance in vibration control among the four dampers. Both the reduction factors of the TLD linearly increase with respect to the mass ratio. This result is consistent with the shake table results of structure– damper systems, such as [3,16]. Most TMDs possess positive vibration reduction effects, with the exception of *R*RMS of the TMD, with a mass ratio of 1%. The *R*RMS of the TMD increases with the increase in mass ratio. According to [6], the mass ratio of the TMD can only reach 0.1%–5% due to installation difficulties and economic costs. These mass ratios are much smaller than the optimal mass ratio, so the limited tuned mass cannot effectively reduce the structural vibrations. This explains the fair performance of the TMDs. A PD with a mass ratio of 3% has a satisfactory performance in terms of both *R*max and *R*RMS, whereas it does not provide a positive control effect for other mass ratios. The mass ratio has negligible influence on the reduction effects of the PTMD. The *R*max of the PTMDs is positive, whereas the *R*RMS is negative. Figure 11 shows that the damper forces have a positive correlation with the mass ratios. In addition, the damper forces of the PDs have a magnitude of 10<sup>6</sup> N, which is much larger than those of the counterpart TLDs. This indicates that a larger damper force cannot ensure better control performance. In sum, for the particular structure and ground motion in this study, these dampers are not always effective at controlling the seismic responses of the four-story steel frame. According to previous studies [3,6–10], the control performance of dampers is influenced by many factors, such as the dynamic characteristics of the primary structure, the frequency characteristics and intensities of the seismic inputs, and the damper parameters. Therefore, the conclusions drawn from Figures 10–12 are specific and not generalizable.

### 5.2.2. Effects of the Integration Parameters of GCR Algorithms

As mentioned in Section 2, the integration parameters, i.e., *κ*1 and *κ*2, of the GCR algorithms may greatly influence the numerical properties of the algorithms. Therefore, the influences of the two integration coefficients on SSTT are investigated. GCR algorithms with four sets of [*<sup>κ</sup>*1, *<sup>κ</sup>*2]=[1/2, 1/2], [1/2, 1], [1, 1/2], [1, 1] are selected, whereas GCR algorithms with *κ*1 = 1/2, *κ*2 = 1/4 are used as the reference model. A time step of 0.001 s is adopted for the integration algorithms. The structural responses of the steel frame with four dampers with a mass ratio of 1% are considered. Figure 13 depicts the lateral displacement at the fourth story using GCR algorithms with different sets of integration coefficients. Table 4 further provides the error indices of the structural responses.

**Figure 13.** Effects of the integration parameters on structural responses. (**a**) TMD; (**b**) TLD; (**c**) PD; (**d**) PTMD.


**Table 4.** Error indices of top lateral displacements with different integration parameters (unit: %).

Figure 13 and Table 4 show that the error indices of the PDs are generally larger than their counterpart TMDs, TLDs, and PTMDs, particularly when *κ*1 = 1/2, *κ*2 = 1/2 and *κ*1 = 1/2, *κ*2 = 1. In addition, the error indices of *κ*1 = 1 normally exceed those of *κ*1 = 1/2 because there exists numerical damping for GCR algorithms with *κ*1 = 1, as shown in Figure 1. However, all the error indices are relatively small (the maximum

NEE and NRMSE are less than 3%). This can be explained as follows: the fundamental frequency of the frame structure *f*1 is 0.98 Hz, thus the corresponding Ω = 2*π f*1Δ*<sup>t</sup>* = 0.006. As the value of Ω is small, the PE and equivalent damping ratio, as indicated in Figure 1, of the integration algorithm is very small. For instance, when *κ*1 = 1, *κ*2 = 1, the PE and equivalent damping ratios of the GCR algorithms are only 1.2 × 10−<sup>5</sup> and 0.0015, respectively. The relatively low error of the integration algorithm leads to small errors in the SSTT results. Therefore, the influences of the integration parameters of the GCR algorithms on SSTT of the frame structure with dampers can be neglected. It should be noted, however, that if a larger time step and a stiffer structure with a larger frequency are adopted, the influences of the integration parameters may be significant.
