## 2 Materials and Methods  
The study focused on the seismic response of buildings with 5, 10, 15, and 20 stories. The detailed multi degree-of-freedom numerical model of each structure, with the mass of each story lumped at the floor level (Figure 1), was created based on the structural parameters used in [33]. Additionally, the bi-linear model of isolation devices (Figure 1), in the form of high-damping rubber bearings, was employed [34]. Specifications of models of buildings equipped with seismic isolation devices are shown in Table 1 (compare [33,34]). The model of the base isolation system is characterized by the following parameters: effective stiffness (\(K_{\text{eff}}\)), effective damping (\(C_{\text{eff}}\)), initial stiffness (\(K_{\text{u}}\)), yield force (\(F_{\text{y}}\)), design displacement (\(D_{\text{D}}\)) and yield displacement (\(D_{\text{y}}\)), and effective period (\(T_{\text{eff}}\)). The non-traditional TMDs were considered in the analysis. These control devices are characterized by a significant reduction of the stroke length, in comparison with the traditional TMDs [33]. As a result of previous analyses, the non-traditional TMDs were proposed and confirmed as much better cooperators with base-isolated structures in hybrid control systems [33]. It should be added that the effectiveness of the devices depends on how much of the passive force is added, compared to the effect of seismic isolators on which the building is resting. Thus, the effectiveness of the added TMDs can be somehow modified by changing the proportion of their contribution. A different number of non-traditional TMDs (1, 4, and 8) was considered in the study. Each TMD was modelled as the single degree-of-freedom system with spring connected to the base of building and dashpot directly connected to the ground (see [33] for details). It should be underlined that such a location of TMDs does not require any special arrangements [33]. Specifications of non-traditional TMDs for different computational cases are presented in Table 2. The TMD models are defined by three parameters: mass (\(M_{\text{TMD}}\)), stiffness (\(K_{\text{TMD}}\)), and damping (\(C_{\text{TMD}}\)).  
It should be underlined that the analyzed structures were designed based on the provisions of several standards. The Iranian National Building Code [35] and the Iranian code of practice for seismic resistance design of buildings (Standard no. 2800) [36] were used for gravity and seismic loads. Moreover, Standard no. 2800 and the Uniform Building Code, 1994 Edition (UBC-94) [37] were applied for the calculation and equalization of seismic parameters. Finally, the structural elements were analyzed using the Load and Resistance Factor Design (LRFD) method, in accordance with IBC 2006 [38] recommendations. Satisfying the requirements for two categories of the limit states (strength and serviceability) has unequivocally confirmed that the structural elements of buildings were correctly designed.  
The numerical investigations were carried out using SAP2000 computational software (CSI, Computers and Structures Inc., v10, CSI, Berkeley, CA, USA). In order to conduct the non-linear time history dynamic analyses, 14 ground motion records were utilized (Table 3). For comparison purposes, all records were scaled to attain the peak acceleration value of 0.5 g, which resulted in the peak ground velocity variation from 13.67 to 95.38 cm/s. The seismic response was calculated in accordance with the time-stepping Newmark method ([39]). Based on the preliminary computations, the time step of 0.01 s was found to be small enough to satisfy the numerical stability and accuracy conditions during the conducted analyses. Therefore, this time step was applied in the time history dynamic analysis.