# Improvement of Performance Level of Steel Moment-Resisting Frames Using Tuned Mass Damper System
## 7 Results and Discussion
### The Results for the Artificial Earthquake
\begin{array}{llll}\mathit{Find}:&M_{d},\;K_{d},\;C_{d}\\ \mathit{Mininize}:&\max\left(\frac{\max\left|\mathit{drift}_{t}\right|_{\mathit{ inthd}}\mathit{TAD}}{\max\left|\mathit{drift}_{t}\right|_{\mathit{ inthd}}\mathit{TAD}}\right)\times 100,&i=1,2,3,\ldots,n&(\mathrm{O}\cdot\mathrm{F}\,1)\\ \mathit{Mininize}:&\max\left(\frac{\max\left|\mathit{Acc_{\mathit{ref}}}\right| _{\mathit{auth}}\mathit{TAD}}{\max\left|\mathit{Acc_{\mathit{ref}}}\right| _{\mathit{auth}}\mathit{TAD}}\right)\times 100,&i=1,2,3,\ldots,n&(\mathrm{O}\cdot \mathrm{F}\,2)\\ \mathit{Mininize}:&\max\left(\frac{\max\left|\mathit{Dispr_{\mathit{ref}}} \right|_{\mathit{auth}}\mathit{TAD}}{\max\left|\mathit{Dispr_{\mathit{ref}}} \right|_{\mathit{auth}}\mathit{TAD}}\right)\times 100,&i=1,2,3,\ldots,n&(\mathrm{O}\cdot \mathrm{F}\,3)\\ \end{array}  \tag{25}\]  
where O.F., \(\mathit{Acc_{\mathit{ref}}}\), and \(\mathit{Dispr_{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{  \mathit{  \mathitmathitmathit{    }}}}}}}}}}}}}}}}\) show the objective function, roof acceleration, and displacement, respectively. The last-second objective functions are usually used by researchers to optimize the parameters of vibration control systems [18; 34; 37; 40]. Additionally, the computational workload, in the form of required running time, for each objective function is presented in Table 5. The calculations have been run on a computer with a 64-bit operating system and CPU Intel(R) Core(TM) i3-4170 CPU 3.70 GHz with 4 GB RAM. \begin{table \begin{tabular}{c c c c c} \hline \hline
**O.F** & \(K_{\mathit{tmd}}\) **(N/mm)** & \(C_{\mathit{tmd}}\) **(N\(\cdot\)s/mm)** & \(M_{\mathit{tmd}}\) **(N\(\cdot\)s\({}^{2}\)/mm)** & **Required Running Time** \\  & & & & **(s)** \\ \hline
**O.F 1** & 405.47 & 935.66 & 180 & 997.854 \\
**O.F. 2** & 211.021 & 1130.6 & 122.87 & 865.528 \\
**O.F. 3** & 250.025 & 749.317 & 163.156 & 1114.548 \\ \hline \hline \end{tabular} 
\end{table}
Table 5: Optimum parameters and required running time achieved for the TMD subjected the artificial earthquake for different objective functions. Figure 14: The performance level of the frame subjected to the artificial earthquake. It can be seen from Table 5 that the required running time for O.F. 2 is shorter than for two other objective functions. Moreover, the computational workload of the proposed objective function (O.F 1) is between those of the two traditional objective functions (O.F. 2 and O.F. 3). Table 6 presents the maximum responses of the controlled and uncontrolled structures subjected to the artificial earthquake for different objective functions. The maximum roof displacement, roof acceleration, and drift ratio for the uncontrolled structures are 0.642 m, 18.06 m/s\({}^{2}\), and 4.09%, respectively. Table 6 shows that O.F 1 has the best performance between all studied objective functions in reducing the maximum responses of the structure and therefore improving the seismic behavior of the structures. Additionally, to evaluate the capability of the method presented in this paper, the 10-story steel moment-resisting frame structure is considered here that has been studied by [PERSON] and [PERSON] [64]. The mass of all stories and the damping ratio for all 10 modes are assumed to be equal to \(m=\) 218,900 kg, and 3%, respectively [64]. The cross-sections and lengths for the beams and columns of the structure are shown in Figure 15. The material has the yield stress equal to 248.2 MPa. Additionally, a gravity uniformly distributed load, equal to 21.89 kN/m, is applied to all beams. The modulus of elasticity is considered to be equal to 200 GPa. All beam-to-column connections are considered to be rigid [64]. The first natural frequency of the 10-story steel frame is 4.19 rad/s, which is equal to the results presented by [PERSON] and [PERSON] in [64]. It can be seen that the differences are negligible, therefore, it confirms the accuracy of the modeling approach.