# Shared Tuned Mass Dampers for Mitigation of Seismic Pounding
## 3 Numerical Study
### Examination with MDOF Systems  
Here we examine the results and conclusions obtained for two SDOF buildings with MDOF models of two buildings. The selected buildings are the same as those used by [PERSON] et al. [28]. Building A has a mass of \(3.1752\times 10^{5}\) kg, and stiffness of \(3.404\times 10^{8}\) N/m on each floor. Building B has a mass of \(3.6287\times 10^{5}\) kg and stiffness of \(6.127\times 10^{8}\) on each floor. Both buildings have 8 floors with a damping coefficient of 100 Ns/m on each floor. Fundamental frequencies of buildings A and B are 6.04, and 7.58 rad/s, respectively. The mass of the TMD is 54,430 kg. [PERSON] et al. [28] present two cases as optimal solutions. Case 1 corresponds to attaching the TMD mass to building A by a dashpot and building B by a spring. Case 2 corresponds to a spring connection with building A and dashpot connection to building B. Case 2 was found to be more effective and is considered here for further analysis. [PERSON] et al. [28] arrived at this solution by specifying weights equal to 1 for the displacements in the top four floors of building A and 0 for all other displacements and velocities in defining the \(\mathbf{Q}\) matrix  
The cost function for this example is presented in Figure 5. The frequency ratio, for this system, is defined as the frequency of the STMD divided by the fundamental frequency of building A. Like the cost function for SDOF buildings shown in Figure 2, the cost function of these MDOF buildings sharing a TMD displays two minima for each damping ratio. The first minima, which is larger than the second one, occurs near the resonance of STMD mass with the first mode of building A. The second minimum occurs at a much larger frequency ratio. The optimal stiffness presented by [PERSON] et al. (2001) is \(1.786\times 10^{8}\) N/m, which is about 50% of the stiffness of each of the floors of building A. Based on this stiffness, the frequency of STMD is 57.28 rad/s, which corresponds to a frequency ratio of 9.48. The optimal damping coefficient reported by [PERSON] et al. [28] is 1.809E6, which, for this frequency ratio corresponds to a damping ratio of 0.29 for the STMD. It is then clear that the optimal solution reported in [PERSON] et al. [28] corresponds to the global minima of the cost function (see purple line in Figure 4).  
Figure 5: The cost function for the two eight-storey buildings presented in [PERSON] et al. [28] as a function of frequency and damping ratios.  
Variation of the peak displacements of buildings A and B with frequency ratio and damping ratio is shown in Figure 6. The solution considered optimal by [PERSON] et al. [28] is indicated in the figure with red dots. The response of building A, to which the STMD mass is connected by a spring, is optimally controlled when the frequency ratio is close to 1, and the damping ratio is low. This solution, however, is not effective in controlling the response of building B. The optimal solution, based on the global minima of \(J\), as denoted by red dots, is effective in controlling bounding as well as reducing the response of both the buildings. However, the solution uses a very stiff spring, and the TMD mass does not move relative to building A.  
This can be confirmed from the low stroke of the device as shown in Figure 6. The STMD mass, in the solution presented by [PERSON] et al. [28] is essentially motionless and does not function as intended. The solution, therefore, is not a tuned mass damper as its frequency is much larger than the frequency of the buildings it is attached to. In fact, it can be verified that the same response reduction can be achieved by coupling the two buildings with a dashpot. If we consider the optimal solution as the one near resonance of STMD with building A, the STMD is not very effective in controlling bounding between the buildings.