## 2 Materials and Methods
### BIR Kinematic Device  
The BIKD used in the modeling, links the superstructure with the substructure through a pivoted mechanism at its base (Figure 4). This device is vertically very rigid and the rotation with respect to its support at the base allows lateral displacement relative to the ground of the superstructure that rolls on it. It has an axially inextensible cable, which links the elastic element inside the isolator with the superstructure through a post-tensioning \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Range** \\ \hline \(H\) & Height between pallets (m) & 1–1.9 \\ \(B\) & Rack length in direction parallel to the aisle (m) & 2–3 \\ \(t\) & Short direction beam length (m) & 0.85 \\ \(N_{sl}\) & Number of storage levels & 2–6 \\ \(T_{iso}\) & Isolation period (s) & 3–5 \\ \(H_{iso}\) & Isolator height (m) & 0.3–0.6 \\ \(\mu\) & Coefficient of friction between materials in the BIKD ’s ball joint & 0.4–0.6 \\ \(d\) & Diameter of the ball joint at the base support of the BIKD (m) & 0.06 \\ \hline \end{tabular}
\end{table} 
Table 2: Range of design parameters used in the analysis with LSH model. load and helps prevent it from up-lifting. The friction in the ball joint at the base of the BIKD's support allows energy dissipation, due to the work between rotation and friction, which is proportional to the weight of the structure. The energy dissipation depends on the diameter of that ball joint and the type of materials in contact, since they modify the coefficient of friction and its effect on the lateral balance of the isolator [13]. The Figure 4 shows the diagram of the isolator, used with a compressed pneumatic piston as the elastic element. It is important to note that in this figure, two cylinders with different diameters are considered; this is not the case of this investigation where a cylinder with a constant diameter was considered, i.e., \(D_{\alpha}^{(1)}=D_{\alpha c}^{(2)}\) in Figure 4.  
In this research, the condition in which the isolator's radius of curvature is constant and equal to its height was used (\(H=R\) in Figure 4a). In this way, the weight of the structure did not interact with the lateral stiffness of the BIKD, since the vertical projection of the contact point between the isolator and the superstructure passed through the center of the ball joint at the basal pivot (Figure 4b). Due to the above, the weight of the structure did not affect the lateral balance of the BIKD, because the eccentricity of that force was zero. Figure 4: Scheme of the BIKD used with a compressed pneumatic piston as elastic element (**a**) undeformed equilibrium position, (**b**) deformed position with maximum displacement, (**c**) Three-dimensional scheme of the finite elements model of the device, and (**d**) Installation diagram of a rack on the isolation system. The figures (**a,b**) were adapted from [14]. Figure 3: Basic structuring of racks with the different storage levels analyzed (from 2 up to 6). Therefore, the lateral stiffness was determined only by the tension of the post-tensioned internal element and its stiffness [13]. [PERSON] et al. [13] have presented a detailed description of the conceptual development of the BIKD and the equations that govern its mechanical behavior. In the present paper, the authors describe the equations under the conditions in which the radius of curvature (\(R\)) is constant and equal to its height (\(H_{iso}\)).