## 4 Numerical Investigation
### Evaluation Criteria  
The comparison of the different control strategies is based on the stationary response properties of the different control structures. From the response time histories, the rms and peak accelerations and displacements at different storeys were obtained. From the obtained values, 12 performance criteria were identified. The first criterion, \(J_{1}\), appraises the ability of the control strategy to reduce rms accelerations:  
\[J_{1}=\max(\sigma_{\bar{x}1},\sigma_{\bar{x}30},\sigma_{\bar{x}50},\sigma_{ \bar{x}55},\sigma_{\bar{x}60},\sigma_{\bar{x}65},\sigma_{\bar{x}70},\sigma_{ \bar{x}75})/\sigma_{\bar{x}750} \tag{14}\]  
where \(\sigma_{\bar{x}i}\) is the rms acceleration of the \(i\)th storey and \(\sigma_{\bar{x}750}\) is the rms acceleration of the 75 th floor (last occupied floor) without control. The second performance criterion evaluates the average performance of six floors above the 49 th floor:  
\[J_{2}=\frac{1}{6}\sum_{i=1}\left(\sigma_{\bar{x}i}/\sigma_{\bar{x}i0}\right) \tag{15}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Strategy** & \begin{tabular}{c} **Max Damping** \\ **Coefficient (kNs/m)** \\ \end{tabular} & \
\begin{tabular}{c} **Min Damping** \\ **Coefficient (kNs/m)** \\ \end{tabular} &
\begin{tabular}{c} **Equivalent** \\ **Damping Ratio** \\ \end{tabular} \\ \hline TMD & 47 & 47 & 4.7\% \\ STMD & 163.4 & 2.61 & 16\%–2.6\% \\ ATMD & 100 & 100 & 10\% \\ SHMD & 168 & 39 & 16.8\%–4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Damping coefficients. For clarity i) TMD, ii) STMD, iii) ATMD, iv) SHMD stand for i) tuned, ii) semi-active tuned, iii) active-tuned and iv) semi-active hybrid mass damper. Figure 11: Ensemble of all the different control options (a) TMD; (b) STMD; (c) ATMD; (d) SHMD studied herein for the model 76-storey structure of [PERSON] et al. [3]. For \(i=50,55,60,65,70,75\); where, \(\sigma_{ziio}\) is the rms of the \(i\)th floor without control. The third and fourth performance indices assess the ability of the control system to reduce top floor displacements:  
\[J_{3}=\sigma_{x76}/\sigma_{x76o} \tag{16}\]  
\[J_{4}=\frac{1}{7}{\sum\limits_{i}}\left(\sigma_{xi}/\sigma_{xiio}\right) \tag{17}\]  
For \(i=50,55,60,65,70,75,76\); where, \(\sigma_{xi}\) is the rms displacement of the \(i\)th floor, \(\sigma_{xiio}\) is the rms displacement of the \(i\)th storey without control and \(\sigma_{x76o}\) is 10.136 cm. The fifth and sixth indices take into account the rms stroke of the damper (i.e., \(i=77\)) and the average power respectively:  
\[J_{5}=\sigma_{x77}/\sigma_{x76o} \tag{18}\]  
\[J_{6}=\left\{\frac{1}{T}\int\limits_{0}^{T}\left[\dot{x}_{77}(t)u(t)\right]^{ 2}dt\right\}^{1/2} \tag{19}\]  
In which, \(\sigma_{x77}\) is the rms stroke of the damper, \(\dot{x}_{77}(t)\) is the damper velocity and \(T\) is the total time of integration. Similarly to the first performance indices, the next four criteria (i.e., \(J_{7}\) to \(J_{10}\)) evaluate the performance in terms of peak response quantities:  
\[J_{7}=\max(\ddot{x}_{p1},\ddot{x}_{p30},\ddot{x}_{p50},\ddot{x}_{p55},\ddot{x} _{p60},\ddot{x}_{p65},\ddot{x}_{p70},\ddot{x}_{p75})/\ddot{x}_{p750} \tag{20}\]  
\[J_{8}=\frac{1}{6}{\sum\limits_{i}}\left(\ddot{x}_{pi}/\ddot{x}_{piio}\right) \tag{21}\]  
For \(i=50,55,60,65,70,75\);  
\[J_{9}=x_{p76}/x_{p76o} \tag{22}\]  
\[J_{10}=\frac{1}{7}{\sum\limits_{i}}\left(x_{pi}/x_{pio}\right) \tag{23}\]  
For \(i=50,55,60,65,70,75,76\); where, \(\ddot{x}_{pi}\) is the peak acceleration of the \(i\)th floor with control and \(\ddot{x}_{pi}\) is the peak acceleration of the \(i\)th floor without control. Similarly, \(x_{pi}\) is the peak displacement of the \(i\)th floor and \(x_{pi}\) is the peak displacement of the \(i\)th floor without control and \(x_{p76o}=32.3\) cm. The \(11\)th criterion assesses the ability of the control strategy to minimise the stroke of the damper:  
Figure 11:0  
In which \(x_{p77}\) is the peak stroke of the actuator.