5.2.3. Effects of the Time Step

The time step is a crucial factor of the integration algorithms and SSTT. A larger time step means saving more computing time while reducing accuracy. It is well known that SSTT requires high computational efficiency for integration algorithms. If an integration algorithm maintains a relatively high accuracy for a larger time step, it is definitely a promising choice for the application of SSTT. Therefore, we studied the influence of the time steps on SSTT. GCR algorithms with *κ*1 = 1/2, *κ*2 = 1/4 and four time steps (Δ*t* = 0.002 s, 0.005 s, 0.01s, 0.02 s) are adopted. The results for GCR algorithms with *κ*1 = 1/2, *κ*2 = 1/4 and a time step of Δ*t* = 0.001 s are used for comparison. The lateral displacements at the top story and the steel frame attached to the four types of dampers with a mass ratio of 1% are provided in Figure 14. The corresponding error indices are tabulated in Table 5.

**Figure 14.** Effects of the time step on structural responses. (**a**) TMD; (**b**) TLD; (**c**) PD; (**d**) PTMD.


**Table 5.** Error indices of top lateral displacements with different time steps (unit: %).

Figure 14 and Table 5 indicate that with the increase in the time step, the error indices for all damper cases increase. Regarding the TMDs and TLDs, even when the time step is very large, i.e., Δ*t* = 0.02 s, the error indices are less than 5%. However, for the PDs and PTMDs, the error indices for Δ*t* = 0.02 s are relatively large and almost reach 30%. The TMD and PD cases have the smallest and largest errors, respectively. The error indices of the PTMD cases are between those of the TMD cases and the PD cases, because the PTMD is a combination of the TMD and PD. A possible reason for the large errors of the PD cases is that the impulsive force induced by the PD has a negative impact on the integration algorithms.
