# Shared Tuned Mass Dampers for Mitigation of Seismic Pounding
## 3 Numerical Study
### An Optimization Example  
An example of the optimization problem is presented here. The properties of buildings A and B are fixed for numerical simulations. Their frequencies are taken as 6.04 and 7.58 rad/s, and the damping ratio is 5% for both. The frequencies are the same as those of the two buildings studied by [PERSON] et al. [28]. The mass of the two buildings are equal, and the mass of the TMD is expressed as a fraction of the mass of building A, i.e., \(m=\rho_{A}m_{A}\). The frequency of the TMD is denoted by \(\omega=\sqrt{\frac{k}{m}}\)  
Figure 1: Schematic representation of a tuned mass damper (TMD) shared by two buildings modeled as single degree of freedom (SDOF) systems. and the frequency ratio is defined as \(f=\frac{\omega}{\omega_{A}}\). The damping ratio of the TMD, expressed in terms of the mass and frequency of the TMD, is denoted by \(\zeta\). The variables of interest are therefore \(f\) and \(\zeta\). Figure 2 shows the variation of the cost function with the frequency ratio and damping ratio of the TMD. The cost function has two minima, which correspond to two solutions. There is a local minimum near a frequency ratio of 1. This minimum is more prominent for lower damping ratios. The global minima occur at much larger frequency ratios, ranging from about 10 to about 90 for damping ratios from 0.4 to 0.05. It is interesting to note that the amplitudes of the global minima are not very sensitive to damping ratios of the TMD. It is then interesting to investigate whether the solution that minimizes \(J\) corresponds to minimum displacements of the two buildings and minimizes the separation distance required to prevent bounding. This distance is the largest negative relative displacement between the two buildings and is hereafter referred to as the bounding distance. To investigate this, we use the north-south component of the El Centro ground motion from the 1940 Imperial Valley Earthquake and simulate the response of the two buildings and STMD with varying frequency and damping ratios. The results are shown in Figure 3. The maximum displacement of the two buildings are shown on the top panel. The maximum stroke of the TMD is shown in the bottom left panel. Here, stroke is defined as the relative displacement of the TMD mass from building A. Pounding distance is shown in the bottom right panel. For comparison, the response of uncontrolled buildings is shown with black dashed lines. Another scenario considered is where an optimally tuned (based on Sadek, [30]) TMD is placed on building A whereas building B is uncontrolled. In this case, the frequency tuning ratio is 0.96 and the damping ratio is 0.22. The results corresponding to this are shown with dashed blue lines. The results show that the local minima, similar to those observed in Figure 2, occurs in the displacement response of building A when the TMD is tuned to the frequency of the building. However, the combination of \(f\) and \(\zeta\) that minimizes \(J\) does not necessarily minimize the displacement of building A. For example, for \(f\) approximately equal to 1, a damping ratio of 10% is required to minimize \(J\), whereas 5% damping ratio minimizes the peak displacement of building A. This can be explained by the larger stroke of the device when the damping ratio is low, as is clearly seen in the bottom right panel of Figure 3. The optimal solution near \(f\approx 1\) is not effective in controlling the peak response of building B.  
Figure 2: Variation of the cost function with frequency ratio and damping ratio of the TMD. Themass of the TMD is taken as 3% of the mass of building A in this example. It is also interesting to note that the locally optimal solution (\(f\approx 1\)) of the shared TMD is more effective in controlling peak response of building A, and with a much smaller damping ratio than when the TMD is placed only on building A. This indicates that sharing a TMD with an adjacent building can increase its effectiveness without adversely affecting the response of the adjacent building.